Recursive Sequences

• a(n) = d * a(n-1) - a(n-2).
• a(n) = d * a(n-1) + d * a(n-2).
• a(n) = a(n-1) + a(n-2).
• a(n) = d * a(n-1) + a(n-2).
• a(n) = d * a(n-1). Geometric progressions.
• a(n) = a(n-1) + d. Arithmetic progressions.
• a(n) = a(n-1). Constants.

This page is synchronized with OEIS in January 2007. Sequences in OEIS may start with a different index.
There is a separate page with proofs. Sequences from the OEIS that pass the recursion test, but are not defined as recursions, are collected on a separate page: Non Recursions.

Common properties:

• The number of words of length n in alphabet {1,2,...,d} avoiding words "11", "22", ..., "kk" is a recurrence sequence with initial terms a(0) = 1, a(1) = d and a recurrence relation a(n) = (d-1)a(n-1) + (d-k)a(n-2). Proof.
• The generating function f(x) = (mx+n)/(1 - dx - kx²) generates a sequence with the recurrence relation a(n) = da(n-1) + ka(n-2) and initial conditions a(0) = n and a(1) = nd + m.
• Let p and q be the roots of the equation x² - dx - k = 0. Then the sequence a(n) = pⁿ + qⁿ = ((d+sqrt(d²+4k))/2)ⁿ + ((d-sqrt(d²+4k))/2)ⁿ satisfies the recurrence a(n) = da(n-1) + ka(n-2) with the initial conditions a(0) = 2, a(1) = d. Proof.
• Let p and q be the roots of the equation x² - dx - k = 0. Then the sequence a(n) = (pⁿ - qⁿ)/(p-q) satisfies the recurrence a(n) = da(n-1) + ka(n-2) with the initial conditions a(0) = 0, a(1) = 1. Proof.

a(n) = d * a(n-1) - a(n-2).

Common properties for sequences with initial terms a(0) = 1, a(1) = d-1:

• a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,2, ..., d-1} which do not end in 0. Proof.
• a(n) equals the number of domino tilings in S_d-1 × P_2n (product of a star graph and a path graph). Proof.
• The generating function of this sequence is (1-x)/(1-dx+x²).
• a(n+1)a(n-1) - a(n)² = d-2. Proof.
• a(n+2)a(n-1) - a(n)a(n+1) = d(d-2). Proof.
• a(n) = (d-1) * a(n-1) + (d-2) * (a(n-2) + a(n-3) + ... + a(1) + a(0)). Proof.

Common properties for sequences with initial terms a(0) = 1, a(1) = d:

• The difference sequence follows the same recursion and is similar (shifted) to the sequence starting with a(0) = 1, a(1) = d-1.
• a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,2, ..., d-1}. Proof.
• a(n) equals the number of domino tilings in S_d-1 × P_2n+1 (product of a star graph and a path graph) with d-3 non central vertices removed from the last star. Proof.
• The generating function of this sequence is 1/(1-dx+x²).
• a(n) = (pⁿ - qⁿ)/(p-q), where p and q are the roots of the equation: x² - dx + 1 = 0.
• a(n+1)a(n-1) - a(n)² = -1. Proof.
• a(n+2)a(n-1) - a(n)a(n+1) = -d. Proof.

Common properties for sequences with initial terms a(0) = 2, a(1) = d:

• The generating function of this sequence is (2-dx)/(1-dx+x²).
• a(n) = pⁿ + qⁿ, where p and q are the roots of the equation: x² - dx + 1 = 0. That is a(n) = ((d+sqrt(d²-4))/2)ⁿ + ((d-sqrt(d²-4))/2)ⁿ. Proof.
• Asymptotically a(n) = Round(pⁿ), where p is the largest root of the equation: x² - dx + 1 = 0. That is a(n) approaches ((d+sqrt(d²-4))/2)ⁿ. Proof.

Sequences:

• a(n) = - 14a(n-1) - a(n-2).
  • Sequence: 1, 1, -15, 209, -2911, 40545, -564719,
    In OEIS: - A122572 a(1)=a(2)=1, a(n)=-14a(n-1)-a(n-2)
• a(n) = - 3a(n-1) - a(n-2).
  • Sequence: 1, -1, 1, -1, 1, -1, 1, -1,
    In OEIS: - A033999 (-1)^n.
  • Sequence: 1, -2, 5, -13, 34, -89, 233, -610,
    In OEIS: - A099496 (-1)^nFib(2n+1).
  • Sequence: -3, 11, -30, 79, -207, 542, -1419, 3715,
    In OEIS: - A098150 a(n) = 2(a(n-2) - a(n-1)) + a(n-3) where a(0)=-3, a(1)=11 & a(2)=-30.
  • Sequence: -1, -1, 4, -11, 29, -76, 199, -521, 1364,
    In OEIS: - A098149 a(0)=-1, a(1)=-1, a(n)=-3*a(n-1)-a(n-2) for n > 1.
• a(n) = - 2a(n-1) - a(n-2).
  • Sequence: 1, -1, 1, -1, 1, -1, 1, -1,
    In OEIS: - A033999 (-1)^n.
  • Sequence: 0, -1, 2, -3, 4, -5, 6, -7, 8, -9,
    In OEIS: - A038608 n*(-1)^n.
• a(n) = - a(n-1) - a(n-2). This sequence is always periodic with period 3.
  • Sequence: 1, 1, -2, 1, 1, -2, 1, 1, -2,
    In OEIS: - A061347 Period 3.
• a(n) = a(n-1) - a(n-2). This sequence is always periodic with period 6.
  • Sequence: 1, 2, 1, -1, -2, -1, 1, 2, 1,
    In OEIS: - A057079 Periodic sequence 1,2,1,-1,-2,-1...; expansion of (1+x)/(1-x+x^2). Also A087204 Periodic sequence: 2,1,-1,-2,-1,1,...
  • Sequence: 1, -2, -3, -1, 2, 3, 1, -2, -3,
    In OEIS: - A117373 Expansion of (1-3x)/(1-x+x^2).
  • Sequence: 1, 3, 2, -1, -3, -2, 1, 3, 2, -1,
    In OEIS: - A119910 Simple periodic sequence with period 1,2,3,-1,-2,-3. submitted definition change
  • Sequence: 1, -3, -4, -1, 3, 4, 1, -3, -4,
    In OEIS: - A117378 Expansion of (1-4x)/(1-x+x^2).
• a(n) = 2a(n-1) - a(n-2). For d = 2 see arithmetic progressions.
• a(n) = 3a(n-1) - a(n-2).
  • Sequence: 1, 2, 5, 13, 34, 89, 233, 610, 1597,
    In OEIS: - A001519 a(n) = F(2n-1) = bisection of Fibonacci sequence. Also A122367 Dimension of 3-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 3 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j). Also A048575 Pisot sequences L(2,5), E(2,5).
  • Sequence: 1, 3, 8, 21, 55, 144, 377, 987, 2584,
    In OEIS: - A001906 F(2n) = bisection of Fibonacci sequence.
  • Sequence: 1, 4, 11, 29, 76, 199, 521, 1364, 3571,
    In OEIS: - A002878 Bisection of Lucas sequence.
  • Sequence: 1, 5, 14, 37, 97, 254, 665, 1741,
    In OEIS: - A054486 A second order recursive sequence.
  • Sequence: 1, 6, 17, 45, 118, 309, 809, 2118,
    In OEIS: - A054492 a(n)=3a(n-1)-a(n-2), a(0)=1,a(0)=6.
  • Sequence: 1, 7, 20, 53, 139, 364, 953, 2495,
    In OEIS: - A055267 a(n)=3a(n-1)-a(n-2); a(0)=1, a(1)=7.
  • Sequence: 1, 8, 23, 61, 160, 419, 1097, 2872,
    In OEIS: - A055273 a(n)=3a(n-1)-a(n-2); a(0)=1, a(1)=8.
  • Sequence: 1, 9, 26, 69, 181, 474, 1241, 3249, 8506,
    In OEIS: - A055849 a(n)=3a(n-1)-a(n-2); a(0)=1, a(1)=9.
  • Sequence: 1, 10, 29, 77, 202, 529, 1385, 3626,
    In OEIS: - A055850 a(n)=3a(n-1)-a(n-2); a(0)=1, a(1)=10.
  • Sequence: 1, 11, 32, 85, 223, 584, 1529, 4003,
    In OEIS: - A056123 a(n)=3a(n-1)-a(n-2); a(0)=1, a(1)=11.
  • Sequence: 2, 3, 7, 18, 47, 123, 322, 843, 2207,
    In OEIS: - A005248 Bisection of Lucas numbers: A000032(2n).
  • Sequence: 2, 6, 16, 42, 110, 288, 754, 1974, 5168,
    In OEIS: - A025169 a(n)=2F(2n+2), where F=A000045 (the Fibonacci sequence).
  • Sequence: 5, 10, 25, 65, 170, 445, 1165, 3050,
    In OEIS: - A106729 Sum of two consecutive squares of Lucas numbers (A001254).
  • Sequence: 7, 19, 50, 131, 343, 898, 2351, 6155,
    In OEIS: - A100545 G.f.: (7-2x)/(x^2-3x+1).
  • Sequence: 11, 15, 34, 87, 227, 594, 1555, 4071,
    In OEIS: - A097512 6*Lucas(2n) - Fib(2n+2).
• a(n) = 4a(n-1) - a(n-2).
  • Sequence: 1, 2, 7, 26, 97, 362, 1351, 5042,
    In OEIS: - A001075 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - a(n-2).
  • Sequence: 1, 3, 11, 41, 153, 571, 2131, 7953,
    In OEIS: - A001835 a(n) = 4a(n-1) - a(n-2); a(0)=a(1)=1. Also A079935 a(n) = 4a(n-1) - a(n-2).
  • Sequence: 1, 4, 15, 56, 209, 780, 2911,
    In OEIS: - A001353 a(n) = 4a(n-1)-a(n-2) with a(0) = 0, a(1) = 1. Also A010905 Pisot sequence E(4,15), a(n)=[ a(n-1)^2/a(n-2)+1/2 ].
  • Sequence: 1, 5, 19, 71, 265, 989, 3691, 13775,
    In OEIS: - A001834 a(0) = 1, a(1) = 5, a(n) = 4a(n-1) - a(n-2).
  • Sequence: 1, 6, 23, 86, 321, 1198, 4471, 16686,
    In OEIS: - A054491 A second order recursive sequence.
  • Sequence: 1, 7, 27, 101, 377, 1407, 5251, 19597,
    In OEIS: - A054485 A second order recursive sequence.
  • Sequence: 1, 8, 31, 116, 433, 1616, 6031, 22508,
    In OEIS: - A055845 a(n)=4a(n-1)-a(n-2); a(0)=1, a(1)=8.
  • Sequence: 2, 4, 14, 52, 194, 724, 2702, 10084,
    In OEIS: - A003500 a(n) = 4a(n-1) - a(n-2).
  • Sequence: 2, 9, 34, 127, 474, 1769, 6602, 24639,
    In OEIS: - A077234 Bisection (odd part) of Chebyshev sequence with diophantine property.
  • Sequence: 3, 9, 33, 123, 459, 1713, 6393, 23859,
    In OEIS: - A082841 a(n)=4a(n-1)-a(n-2).
  • Sequence: 0, 3, 12, 45, 168, 627, 2340, 8733,
    In OEIS: - A005320 a(n) = 4a(n-1) - a(n-2).
  • Sequence: 4, 9, 32, 119, 444, 1657, 6184, 23079,
    In OEIS: - A057819 a(0)=4, a(1)=9, a(n)=4a(n-1)-a(n-2).
  • Sequence: 4, 11, 40, 149, 556, 2075, 7744, 28901,
    In OEIS: - A077236 Bisection (even part) of Chebyshev sequence with diophantine property.
  • Sequence: 0, -1, -4, -15, -56, -209, -780, -2911, -10864,
    In OEIS: - A106707 First entry of the vector (M^n)v, where M is the 2x2 matrix [[0,-1],[1,4]] and v is the column vector [0,1].
  • Sequence: 5, 16, 59, 220, 821, 3064, 11435,
    In OEIS: - A077235 Bisection (odd part) of Chebyshev sequence with diophantine property.
• a(n) = 5a(n-1) - a(n-2).
  • Sequence: 1, 2, 9, 43, 206, 987, 4729, 22658,
    In OEIS: - A002310 a(n) = 5*a(n-1) - a(n-2).
  • Sequence: 1, 3, 14, 67, 321, 1538, 7369, 35307,
    In OEIS: - A002320 a(n) = 5*a(n-1) - a(n-2).
  • Sequence: 1, 4, 19, 91, 436, 2089, 10009, 47956,
    In OEIS: - A004253 a(n) = 5a(n-1) - a(n-2).
  • Sequence: 1, 5, 24, 115, 551, 2640, 12649,
    In OEIS: - A004254 a(n) = 5a(n - 1) - a(n - 2), a(0) = 0, a(1) = 1.
  • Sequence: 1, 6, 29, 139, 666, 3191, 15289,
    In OEIS: - A030221 Chebyshev even indexed U-polynomials evaluated at sqrt(7)/2.
  • Sequence: 1, 7, 34, 163, 781, 3742, 17929, 85903,
    In OEIS: - A055271 a(n)=5a(n-1)-a(n-2); a(0)=1, a(1)=7.
  • Sequence: 1, 9, 44, 211, 1011, 4844, 23209, 111201,
    In OEIS: - A099867 a(n) = 5a(n - 1) - a(n - 2), a(0) = 1, a(1) = 9.
  • Sequence: 1, 13, 64, 307, 1471, 7048, 33769, 161797,
    In OEIS: - A054477 A Pellian-related sequence.
  • Sequence: 2, 5, 23, 110, 527, 2525, 12098, 57965,
    In OEIS: - A003501 a(n) = 5a(n-1) - a(n-2).
  • Sequence: 3, 25, 122, 585, 2803, 13430, 64347,
    In OEIS: - A099868 a(n) = 5a(n - 1) - a(n - 2), a(0) = 3, a(1) = 25.
• a(n) = 6a(n-1) - a(n-2).
  • Sequence: 1, 2, 11, 64, 373, 2174, 12671, 73852, 430441,
    In OEIS: - A038725 A second order recursive sequence.
  • Sequence: 1, 3, 17, 99, 577, 3363, 19601, 114243,
    In OEIS: - A001541 a(0) = 1, a(1) = 3; for n > 1, a(n) = 6a(n-1) - a(n-2).
  • Sequence: 1, 4, 23, 134, 781, 4552, 26531, 154634,
    In OEIS: - A038723 A second order recursive sequence.
  • Sequence: 1, 5, 29, 169, 985, 5741, 33461, 195025,
    In OEIS: - A001653 Numbers n such that 2*n^2 - 1 is a square.
  • Sequence: 1, 6, 35, 204, 1189, 6930, 40391,
    In OEIS: - A001109 a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1.
  • Sequence: 1, 7, 41, 239, 1393, 8119, 47321, 275807,
    In OEIS: - A002315 NSW numbers: a(n) = 6*a(n-1) - a(n-2); also a(n)^2 - 2*b(n)^2 = -1 with b(n)=A001653(n).
  • Sequence: 1, 8, 47, 274, 1597, 9308, 54251, 316198,
    In OEIS: - A054488 A second order recursive sequence.
  • Sequence: 1, 9, 53, 309, 1801, 10497, 61181, 356589,
    In OEIS: - A038761 a(n)=6a(n-1)-a(n-2), n >= 2, a(0)=1, a(1)=9.
  • Sequence: 1, 10, 59, 344, 2005, 11686, 68111, 396980,
    In OEIS: - A054489 A second order recursive sequence.
  • Sequence: 1, 11, 65, 379, 2209, 12875, 75041,
    In OEIS: - A054490 A Pellian-related second order recursive sequence.
  • Sequence: 2, 12, 70, 408, 2378, 13860, 80782,
    In OEIS: - A001542 a(n) = 6a(n-1) - a(n-2).
  • Sequence: 2, 6, 34, 198, 1154, 6726, 39202, 228486,
    In OEIS: - A003499 a(0) = 2, a(1) = 6; for n >= 2, a(n) = 6a(n-1) - a(n-2).
  • Sequence: 2, 10, 58, 338, 1970, 11482, 66922, 390050,
    In OEIS: - A075870 2*n^2 - 4 is a square.
  • Sequence: 2, 13, 76, 443, 2582, 15049, 87712, 511223,
    In OEIS: - A077413 Bisection (odd part) of Chebyshev sequence with diophantine property.
  • Sequence: 2, 14, 82, 478, 2786, 16238, 94642, 551614,
    In OEIS: - A077444 Numbers n such that (n^2 + 4)/2 is a square.
  • Sequence: 3, 9, 51, 297, 1731, 10089, 58803, 342729,
    In OEIS: - A106329 Numbers k such that k^2 = 8*(j^2) + 9.
  • Sequence: 3, 13, 75, 437, 2547, 14845, 86523,
    In OEIS: - A038762 A Pellian-related sequence.
  • Sequence: 3, 15, 87, 507, 2955, 17223, 100383,
    In OEIS: - A075841 2*n^2 - 9 is a square.
  • Sequence: 3, 18, 105, 612, 3567, 20790, 121173,
    In OEIS: - A106328 Numbers j such that 8*(j^2) + 9 = k^2 for some positive number k.
  • Sequence: 4, 20, 116, 676, 3940, 22964, 133844, 780100,
    In OEIS: - A077445 Numbers n such that (n^2 - 8)/2 is a square.
  • Sequence: 4, 22, 128, 746, 4348, 25342, 147704,
    In OEIS: - A100525 Bisection of A048654.
  • Sequence: 0, 4, 24, 140, 816, 4756, 27720, 161564,
    In OEIS: - A005319 a(n) = 6a(n-1) - a(n-2).
  • Sequence: 5, 23, 133, 775, 4517, 26327, 153445,
    In OEIS: - A077240 Bisection (even part) of Chebyshev sequence with diophantine property.
  • Sequence: 5, 27, 157, 915, 5333, 31083, 181165,
    In OEIS: - A101386 G.f.: (5-3x)/(x^2-6x+1).
  • Sequence: 0, 6, 36, 210, 1224, 7134, 41580, 242346,
    In OEIS: - A075848 2*n^2 + 9 is a square.
  • Sequence: 7, 37, 215, 1253, 7303, 42565, 248087,
    In OEIS: - A077239 Bisection (odd part) of Chebyshev sequence with diophantine property.
  • Sequence: 0, 8, 48, 280, 1632, 9512, 55440, 323128,
    In OEIS: - A081554 a(n)=sqrt(2)((3+2sqrt(2))^n-(3-2sqrt(2))^n).
• a(n) = 7a(n-1) - a(n-2).
  • Sequence: 1, 5, 34, 233, 1597, 10946, 75025, 514229,
    In OEIS: - A033889 Fibonacci(4n+1).
  • Sequence: 1, 6, 41, 281, 1926, 13201, 90481, 620166,
    In OEIS: - A049685 a(n)=L(4n+2)/3, where L=A000032 (the Lucas sequence).
  • Sequence: 1, 7, 48, 329, 2255, 15456, 105937,
    In OEIS: - A004187 a(n) = 7*a(n-1) - a(n-2).
  • Sequence: 1, 8, 55, 377, 2584, 17711, 121393,
    In OEIS: - A033890 Fibonacci(4n+2).
  • Sequence: 1, 11, 76, 521, 3571, 24476, 167761,
    In OEIS: - A056914 a(n)=L(4n+1) where L() are the Lucas numbers.
  • Sequence: 2, 7, 47, 322, 2207, 15127, 103682,
    In OEIS: - A056854 a(n)=7a(n-1)-a(n-2), a(0)=2, a(1)=7.
  • Sequence: 2, 13, 89, 610, 4181, 28657, 196418,
    In OEIS: - A033891 Fibonacci(4n+3).
  • Sequence: 3, 21, 144, 987, 6765, 46368, 317811,
    In OEIS: - A033888 Fibonacci(4n).
• a(n) = 8a(n-1) - a(n-2).
  • Sequence: 1, 4, 31, 244, 1921, 15124, 119071,
    In OEIS: - A001091 a(n) = 8a(n-1) - a(n-2); a(0) = 1, a(1) = 4.
  • Sequence: 1, 5, 39, 307, 2417, 19029, 149815,
    In OEIS: - A105426 a(0)=1, a(1)=5, a(n)=8*a(n-1)-a(n-2).
  • Sequence: 1, 7, 55, 433, 3409, 26839, 211303, 1663585,
    In OEIS: - A070997 a(n) = 8*a(n-1) - a(n-2), a(0)=1, a(-1)=1.
  • Sequence: 1, 8, 63, 496, 3905, 30744, 242047,
    In OEIS: - A001090 a(n) = 8*a(n-1)-a(n-2); a(0) = 0, a(1) = 1.
  • Sequence: 1, 9, 71, 559, 4401, 34649, 272791,
    In OEIS: - A057080 Even indexed Chebyshev U-polynomials evaluated at sqrt(10)/2.
  • Sequence: 1, 10, 79, 622, 4897, 38554, 303535,
    In OEIS: - A077245 Bisection (even part) of Chebyshev sequence with diophantine property.
  • Sequence: 2, 8, 62, 488, 3842, 30248, 238142,
    In OEIS: - A086903 a(n) = 8a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 8, a(n) = (4+sqrt(15))^n + (4-sqrt(15))^n.
  • Sequence: 2, 13, 102, 803, 6322, 49773, 391862,
    In OEIS: - A077246 Bisection (even part) of Chebyshev sequence with diophantine property.
  • Sequence: 2, 17, 134, 1055, 8306, 65393, 514838,
    In OEIS: - A077243 Bisection (odd part) of Chebyshev sequence with diophantine property.
  • Sequence: 3, 22, 173, 1362, 10723, 84422, 664653,
    In OEIS: - A077244 Bisection (odd part) of Chebyshev sequence with diophantine property.
• a(n) = 9a(n-1) - a(n-2).
  • Sequence: 1, 8, 71, 631, 5608, 49841, 442961, 3936808,
    In OEIS: - A070998 a(n) = 9*a(n-1) - a(n-2), a(0)=1, a(-1)=1.
  • Sequence: 1, 9, 80, 711, 6319, 56160, 499121,
    In OEIS: - A018913 a(n) = 9a(n - 1) - a(n - 2); a(0) = 0, a(1) = 1.
  • Sequence: 1, 10, 89, 791, 7030, 62479, 555281,
    In OEIS: - A057081 Even indexed Chebyshev U-polynomials evaluated at sqrt(11)/2.
  • Sequence: 2, 9, 79, 702, 6239, 55449, 492802,
    In OEIS: - A056918 a(n)=9*a(n-1)-a(n-2); a(0)=2, a(1)=9.
  • Sequence: 3, 27, 240, 2133, 18957, 168480, 1497363,
    In OEIS: - A065100 a(0) = c, a(1) = p*c^3; a(n+2) = p*c^2*a(n+1) - a(n), for p = 1, c = 3.
• a(n) = 10a(n-1) - a(n-2).
  • Sequence: 0, 2, 20, 198, 1960, 19402, 192060, 1901198,
    In OEIS: - A001078 a(n) = 10*a(n-1)-a(n-2) with a(0) = 0, a(1) = 2.
  • Sequence: 1, 5, 49, 485, 4801, 47525, 470449,
    In OEIS: - A001079 a(n) = 10a(n-1) - a(n-2); a(0) = 1, a(1) = 5.
  • Sequence: 1, 9, 89, 881, 8721, 86329, 854569,
    In OEIS: - A072256 a(n) = 10*a(n-1) - a(n-2); a(0) = a(1) = 1.
  • Sequence: 1, 10, 99, 980, 9701, 96030, 950599,
    In OEIS: - A004189 a(n) = 10*a(n-1)-a(n-2); a(0) = 0, a(1) = 1.
  • Sequence: 1, 11, 109, 1079, 10681, 105731, 1046629,
    In OEIS: - A054320 G.f.: (1+x)/(1-10*x+x^2).
  • Sequence: 1, 12, 119, 1178, 11661, 115432, 1142659,
    In OEIS: - A077251 Bisection (even part) of Chebyshev sequence with diophantine property.
  • Sequence: 2, 10, 98, 970, 9602, 95050, 940898,
    In OEIS: - A087799 a(n) =10a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 10, a(n) = (5+sqrt(24))^n + (5-sqrt(24))^n.
  • Sequence: 2, 21, 208, 2059, 20382, 201761, 1997228,
    In OEIS: - A077249 Bisection (odd part) of Chebyshev sequence with diophantine property.
  • Sequence: 0, 4, 40, 396, 3920, 38804, 384120, 3802396,
    In OEIS: - A122652 a(0)=0, a(1)=4, a(n)=10*a(n-1)-a(n-2).
  • Sequence: 0, 6, 60, 594, 5880, 58206, 576180,
    In OEIS: - A122653 a(0)=0, a(1)=6, a(n)=10*a(n-1)-a(n-2).
  • Sequence: 7, 59, 583, 5771, 57127, 565499, 5597863,
    In OEIS: - A077409 Bisection (even part) of Chebyshev sequence with diophantine property.
  • Sequence: 11, 103, 1019, 10087, 99851, 988423, 9784379,
    In OEIS: - A077250 Bisection (odd part) of Chebyshev sequence with diophantine property.
• a(n) = 11a(n-1) - a(n-2).
  • Sequence: 1, 10, 109, 1189, 12970, 141481, 1543321,
    In OEIS: - A078922 a(n) = 11*a(n-1) - a(n-2).
  • Sequence: 1, 11, 120, 1309, 14279, 155760, 1699081,
    In OEIS: - A004190 Expansion of 1/(1-11*x+x^2).
  • Sequence: 1, 12, 131, 1429, 15588, 170039, 1854841,
    In OEIS: - A097783 Chebyshev polynomials S(n,11) + S(n-1,11) with diophantine property.
  • Sequence: 2, 11, 119, 1298, 14159, 154451, 1684802,
    In OEIS: - A057076 A Chebyshev or generalized Fibonacci sequence.
  • Sequence: 3, 33, 360, 3927, 42837, 467280, 5097243,
    In OEIS: - A075835 13*n^2 + 4 is a square.
• a(n) = 12a(n-1) - a(n-2).
  • Sequence: 1, 6, 71, 846, 10081, 120126, 1431431,
    In OEIS: - A023038 a(n) = 12a(n-1) - a(n-2).
  • Sequence: 1, 11, 131, 1561, 18601, 221651, 2641211,
    In OEIS: - A077417 Chebyshev T-sequence with diophantine property.
  • Sequence: 1, 12, 143, 1704, 20305, 241956, 2883167,
    In OEIS: - A004191 Expansion of 1/(1-12*x+x^2).
  • Sequence: 1, 13, 155, 1847, 22009, 262261, 3125123,
    In OEIS: - A077416 Chebyshev S-sequence with diophantine property.
  • Sequence: 2, 12, 142, 1692, 20162, 240252, 2862862,
    In OEIS: - A087800 a(n) =12a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 12, a(n) = (6+sqrt(35))^n + (6-sqrt(35))^n.
  • Sequence: 2, 24, 286, 3408, 40610, 483912, 5766334,
    In OEIS: - A065101 a(0) = c, a(1) = p*c^3; a(n+2) = p*c^2*a(n+1) - a(n), for p = 3, c = 2.
• a(n) = 13a(n-1) - a(n-2).
  • Sequence: 1, 12, 155, 2003, 25884, 334489, 4322473,
    In OEIS: - A085260 Ratio-determined insertion sequence I(0.0833344) (see the link below).
  • Sequence: 1, 13, 168, 2171, 28055, 362544, 4685017,
    In OEIS: - A078362 A Chebyshev S-sequence with diophantine property.
  • Sequence: 2, 13, 167, 2158, 27887, 360373, 4656962,
    In OEIS: - A078363 A Chebyshev T-sequence with diophantine property.
• a(n) = 14a(n-1) - a(n-2).
  • Sequence: 1, 7, 97, 1351, 18817, 262087, 3650401,
    In OEIS: - A011943 Numbers n such that any group of n consecutive integers has integral standard deviation {viz. A011944(n)}.
  • Sequence: 1, 11, 153, 2131, 29681, 413403,
    In OEIS: - A122769 Numbers n such that n^2 is of the form 1+2m+3m^2 (A056109).
  • Sequence: 1, 13, 181, 2521, 35113, 489061,
    In OEIS: - A001570 Numbers n such that n^2 is simultaneously square and centered hexagonal. Also A122571 a(1)=a(2)=1, a(n)=14a(n-1)-a(n-2).
  • Sequence: 1, 14, 195, 2716, 37829, 526890,
    In OEIS: - A007655 Standard deviation of A007654.
  • Sequence: 1, 15, 209, 2911, 40545, 564719, 7865521,
    In OEIS: - A028230 Bisection of A001353. Indices of square numbers which are also octagonal.
  • Sequence: 2, 2, 26, 362, 5042, 70226, 978122,
    In OEIS: - A094347 a(n) = 14*a(n-1)-a(n-2); a(0) = a(1) = 2.
  • Sequence: 2, 12, 142, 1692, 20162, 240252, 2862862,
    In OEIS: - A087800 a(n) =12a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 12, a(n) = (6+sqrt(35))^n + (6-sqrt(35))^n.
  • Sequence: 2, 14, 194, 2702, 37634, 524174, 7300802,
    In OEIS: - A067902 a(n) = 14*a(n-1) - a(n-2); a(0) = 2, a(1) = 14.
  • Sequence: 0, 2, 28, 390, 5432, 75658, 1053780,
    In OEIS: - A011944 a(n) = 14*a(n-1)-a(n-2) with a(0) = 0, a(1) = 2.
  • Sequence: 0, 6, 84, 1170, 16296, 226974, 3161340,
    In OEIS: - A011945 Area of triangles with integral side lengths a-1, a, a+1 and integral area.
  • Sequence: 0, 8, 112, 1560, 21728, 302632, 4215120,
    In OEIS: - A067900 a(n) = 14*a(n-1) - a(n-2); a(0) = 0, a(1) = 8.
• a(n) = 15a(n-1) - a(n-2).
  • Sequence: 1, 15, 224, 3345, 49951, 745920, 11138849,
    In OEIS: - A078364 A Chebyshev S-sequence with diophantine property.
  • Sequence: 2, 15, 223, 3330, 49727, 742575, 11088898,
    In OEIS: - A078365 A Chebyshev T-sequence with diophantine property.
• a(n) = 16a(n-1) - a(n-2).
  • Sequence: 1, 8, 127, 2024, 32257, 514088, 8193151,
    In OEIS: - A001081 a(n) = 16a(n-1) - a(n-2).
  • Sequence: 1, 16, 255, 4064, 64769, 1032240, 16451071,
    In OEIS: - A077412 Chebyshev U(n,x) polynomial evaluated at x=8.
  • Sequence: 0, 3, 48, 765, 12192, 194307, 3096720,
    In OEIS: - A001080 a(n) = 16*a(n-1)-a(n-2) with a(0) = 0, a(1) = 3.
  • Sequence: 2, 16, 254, 4048, 64514, 1028176, 16386302,
    In OEIS: - A090727 a(n) = 16a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 16.
• a(n) = 17a(n-1) - a(n-2).
  • Sequence: 1, 17, 288, 4879, 82655, 1400256, 23721697,
    In OEIS: - A078366 A Chebyshev S-sequence with diophantine property.
  • Sequence: 2, 17, 287, 4862, 82367, 1395377, 23639042,
    In OEIS: - A078367 A Chebyshev T-sequence with diophantine property.
• a(n) = 18a(n-1) - a(n-2).
  • Sequence: 1, 9, 161, 2889, 51841, 930249, 16692641,
    In OEIS: - A023039 a(n) = 18a(n-1) - a(n-2).
  • Sequence: 1, 17, 305, 5473, 98209, 1762289, 31622993,
    In OEIS: - A007805 a(n)=F(6n+3)/2, where F=A000045 (the Fibonacci sequence).
  • Sequence: 1, 18, 323, 5796, 104005, 1866294, 33489287,
    In OEIS: - A049660 a(n)=F(6n)/8, where F=A000045 (the Fibonacci sequence).
  • Sequence: 1, 19, 341, 6119, 109801, 1970299, 35355581,
    In OEIS: - A049629 a(n)=(F(6n+5)-F(6n+1))/4=(F(6n+4)+F(6n+2))/4, where F=A000045 (the Fibonacci sequence).
  • Sequence: 2, 18, 322, 5778, 103682, 1860498, 33385282,
    In OEIS: - A087215 Lucas(6n): a(n) = 18a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 18.
  • Sequence: 2, 38, 682, 12238, 219602, 3940598, 70711162,
    In OEIS: - A075796 5*x^2 + 5 is a square.
  • Sequence: 3, 51, 915, 16419, 294627, 5286867,
    In OEIS: - A075869 5*n^2 - 9 is a square.
  • Sequence: 3, 54, 969, 17388, 312015, 5598882, 100467861,
    In OEIS: - A065102 a(0) = c, a(1) = p*c^3; a(n+2) = p*c^2*a(n+1) - a(n), for p = 2, c = 3.
  • Sequence: 3, 55, 987, 17711, 317811, 5702887, 102334155,
    In OEIS: - A103134 Fib(6n+4).
  • Sequence: 0, 4, 72, 1292, 23184, 416020, 7465176,
    In OEIS: - A060645 a(0) = 0, a(1) = 4 then a(n) = 18*a(n-1)-a(n-2).
• a(n) = 19a(n-1) - a(n-2).
  • Sequence: 1, 19, 360, 6821, 129239, 2448720, 46396441,
    In OEIS: - A078368 A Chebyshev S-sequence with diophantine property..
  • Sequence: 2, 19, 359, 6802, 128879, 2441899, 46267202,
    In OEIS: - A078369 A Chebyshev T-sequence with diophantine property.
• a(n) = 20a(n-1) - a(n-2).
  • Sequence: 0, 3, 60, 1197, 23880, 476403, 9504180,
    In OEIS: - A001084 a(n) = 20*a(n-1)-a(n-2) with a(0) = 0, a(1) = 3.
  • Sequence: 1, 10, 199, 3970, 79201, 1580050, 31521799,
    In OEIS: - A001085 a(n) = 20a(n-1) - a(n-2).
  • Sequence: 1, 19, 379, 7561, 150841, 3009259, 60034339,
    In OEIS: - A075839 11*n^2 - 2 is a square.
  • Sequence: 1, 20, 399, 7960, 158801, 3168060,
    In OEIS: - A075843 99*a(n)^2 + 1 is a square.
  • Sequence: 1, 21, 419, 8359, 166761, 3326861,
    In OEIS: - A083043 Integers y such that 11x^2-9y^2=2 for some integer x.
  • Sequence: 2, 20, 398, 7940, 158402, 3160100,
    In OEIS: - A090728 a(n) = 20a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 20.
  • Sequence: 0, 6, 120, 2394, 47760, 952806, 19008360,
    In OEIS: - A075844 11*n^2 + 4 is a square.
• a(n) = 21a(n-1) - a(n-2).
  • Sequence: 1, 21, 440, 9219, 193159, 4047120,
    In OEIS: - A092499 Chebyshev polynomials S(n-1,21) with diophantine property.
  • Sequence: 2, 21, 439, 9198, 192719, 4037901,
    In OEIS: - A090729 a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.
• a(n) = 22a(n-1) - a(n-2).
  • Sequence: 1, 11, 241, 5291, 116161, 2550251,
    In OEIS: - A077422 Chebyshev sequence T(n,11) with diophantine property.
  • Sequence: 1, 22, 483, 10604, 232805, 5111106,
    In OEIS: - A077421 Chebyshev sequence U(n,11)=S(n,22) with diophantine property.
  • Sequence: 2, 22, 482, 10582, 232322, 5100502,
    In OEIS: - A090730 a(n) = 22a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 22.
• a(n) = 23a(n-1) - a(n-2).
  • Sequence: 1, 23, 528, 12121, 278255, 6387744,
    In OEIS: - A097778 Chebyshev polynomials S(n,23) with diophantine property.
  • Sequence: 2, 23, 527, 12098, 277727, 6375623,
    In OEIS: - A090731 a(n) = 23a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 23.
• a(n) = 24a(n-1) - a(n-2).
  • Sequence: 1, 12, 287, 6876, 164737, 3946812, 94558751,
    In OEIS: - A077424 Chebyshev sequence T(n,12) with diophantine property.
  • Sequence: 1, 24, 575, 13776, 330049, 7907400,
    In OEIS: - A077423 Chebyshev sequence U(n,12)=S(n,24) with diophantine property.
  • Sequence: 2, 24, 574, 13752, 329474, 7893624,
    In OEIS: - A090732 a(n) = 24a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 24.
• a(n) = 25a(n-1) - a(n-2).
  • Sequence: 1, 25, 624, 15575, 388751, 9703200,
    In OEIS: - A097780 Chebyshev polynomials S(n,25) with diophantine property.
  • Sequence: 2, 25, 623, 15550, 388127, 9687625,
    In OEIS: - A090733 a(n) = 25a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.
• a(n) = 26a(n-1) - a(n-2).
  • Sequence: 1, 13, 337, 8749, 227137, 5896813,
    In OEIS: - A097308 Chebyshev T-polynomials T(n,13) with diophantine property.
  • Sequence: 1, 26, 675, 17524, 454949, 11811150,
    In OEIS: - A097309 Chebyshev polynomials of the second kind, U(n,x), evaluated at x=13.
  • Sequence: 2, 26, 674, 17498, 454274, 11793626,
    In OEIS: - A090247 a(n) = 26a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 26.
• a(n) = 27a(n-1) - a(n-2).
  • Sequence: 1, 26, 701, 18901, 509626, 13741001,
    In OEIS: - A097835 First differences of Chebyshev polynomials S(n,27)=A097781(n) with diophantine property.
  • Sequence: 1, 27, 728, 19629, 529255, 14270256,
    In OEIS: - A097781 Chebyshev polynomials S(n,27) with diophantine property.
  • Sequence: 1, 28, 755, 20357, 548884, 14799511,
    In OEIS: - A097834 Chebyshev polynomials S(n,27) + S(n-1,27) with diophantine property.
  • Sequence: 2, 27, 727, 19602, 528527, 14250627,
    In OEIS: - A090248 a(n) =27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27.
• a(n) = 28a(n-1) - a(n-2).
  • Sequence: 1, 14, 391, 10934, 305761, 8550374,
    In OEIS: - A097310 Chebyshev T-polynomials T(n,14) with diophantine property.
  • Sequence: 1, 28, 783, 21896, 612305, 17122644,
    In OEIS: - A097311 Chebyshev polynomials of the second kind, U(n,x), evaluated at x=14.
  • Sequence: 2, 28, 782, 21868, 611522, 17100748,
    In OEIS: - A090249 a(n) =28a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 28.
• a(n) = 29a(n-1) - a(n-2).
  • Sequence: 1, 29, 840, 24331, 704759, 20413680,
    In OEIS: - A097782 Chebyshev polynomials S(n,29) with diophantine property.
  • Sequence: 2, 29, 839, 24302, 703919, 20389349,
    In OEIS: - A090251 a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29.
• a(n) = 30a(n-1) - a(n-2).
  • Sequence: 1, 15, 449, 13455, 403201, 12082575,
    In OEIS: - A068203 Chebyshev T-polynomials T(n,15) with diophantine property.
  • Sequence: 1, 30, 899, 26940, 807301, 24192090,
    In OEIS: - A097313 Chebyshev polynomials of the second kind, U(n,x), evaluated for x=15.
  • Sequence: 4, 120, 3596, 107760, 3229204, 96768360,
    In OEIS: - A068204 Let (x_n, y_n) be n-th solution to the Pell equation x^2 = 14*y^2 + 1. Sequence gives {y_n}.
• a(n) = 31a(n-1) - a(n-2).
  • Sequence: 1, 30, 929, 28769, 890910, 27589441,
    In OEIS: - A111216 a(n)=31*a(n-1)-a(n-2).
• a(n) = 32a(n-1) - a(n-2).
  • Sequence: 1, 32, 1023, 32704, 1045505, 33423456,
    In OEIS: - A029548 Expansion of 1/(1-32*x+x^2).
• a(n) = 34a(n-1) - a(n-2).
  • Sequence: 1, 17, 577, 19601, 665857, 22619537,
    In OEIS: - A056771 a(n)=a(-n)=34a(n-1)-a(n-2) and a(0)=1.
  • Sequence: 1, 33, 1121, 38081, 1293633, 43945441,
    In OEIS: - A077420 Bisection of Chebyshev sequence T(n,3) (odd part) with diophantine property.
  • Sequence: 1, 34, 1155, 39236, 1332869, 45278310,
    In OEIS: - A029547 Expansion of 1/(1-34*x+x^2). Also A091761 Pell(4n)/Pell(4).
  • Sequence: 1, 35, 1189, 40391, 1372105, 46611179,
    In OEIS: - A046176 Indices of square numbers which are also hexagonal.
  • Sequence: 1, 38, 1291, 43856, 1489813, 50609786,
    In OEIS: - A027657 Expansion of (1+4*x)/(1-34*x+x^2).
  • Sequence: 0, 6, 204, 6930, 235416, 7997214, 271669860,
    In OEIS: - A082405 a(n) = 34*a(n-1) - a(n-2); a(0)=0, a(1)=6.
• a(n) = 38a(n-1) - a(n-2).
  • Sequence: 1, 19, 721, 27379, 1039681, 39480499,
    In OEIS: - A078986 Chebyshev T(n,19) polynomial.
  • Sequence: 1, 37, 1405, 53353, 2026009, 76934989,
    In OEIS: - A097315 Pell equation solutions (3*b(n))^2 - 10*a(n)^2 = -1 with b(n):=A097314(n), n ≥ 0.
  • Sequence: 1, 38, 1443, 54796, 2080805, 79015794,
    In OEIS: - A078987 Chebyshev U(n,x) polynomial evaluated at x=19.
  • Sequence: 1, 39, 1481, 56239, 2135601, 81096599,
    In OEIS: - A097314 Pell equation solutions (3*a(n))^2 - 10*b(n)^2 = -1 with b(n):=A097315(n), n>=0.
  • Sequence: 0, 6, 228, 8658, 328776, 12484830, 474094764,
    In OEIS: - A084070 a(0)=0, a(1)=6, a(n)=38*a(n-1)-a(n-2).
• a(n) = 47a(n-1) - a(n-2).
  • Sequence: 1, 41, 1926, 90481, 4250681, 199691526,
    In OEIS: - A049676 a(n)=(F(8n+3)+F(8n+1))/3, where F=A000045 (the Fibonacci sequence).
  • Sequence: 1, 47, 2208, 103729, 4873055, 228929856,
    In OEIS: - A049668 a(n)=F(8n)/21, where F=A000045 (the Fibonacci sequence).
  • Sequence: 1, 48, 2255, 105937, 4976784, 233802911,
    In OEIS: - A049678 a(n)=F(8n+4)/3, where F=A000045 (the Fibonacci sequence)..
  • Sequence: 2, 47, 2207, 103682, 4870847, 228826127,
    In OEIS: - A087265 Lucas numbers L(8n).
  • Sequence: 3, 137, 6436, 302355, 14204249, 667297348,
    In OEIS: - A049677 a(n)=(F(8n+6)+F(8n+1))/3, where F=A000045 (the Fibonacci sequence).
  • Sequence: 6, 281, 13201, 620166, 29134601, 1368706081,
    In OEIS: - A049679 a(n)=(F(8n+7)+F(8n+5))/3, where F=A000045 (the Fibonacci sequence).
  • Sequence: 0, 7, 329, 15456, 726103, 34111385,
    In OEIS: - A049686 a(n)=F(8n)/3, where F=A000045 (the Fibonacci sequence).
• a(n) = 48a(n-1) - a(n-2).
  • Sequence: 1, 24, 1151, 55224, 2649601, 127125624,
    In OEIS: - A114051 x such that x^2 - 23*y^2 = 1.
• a(n) = 51a(n-1) - a(n-2).
  • Sequence: 1, 50, 2549, 129949, 6624850, 337737401,
    In OEIS: - A097838 First differences of Chebyshev polynomials S(n,51)=A097836(n) with diophantine property.
  • Sequence: 1, 51, 2600, 132549, 6757399, 344494800,
    In OEIS: - A097836 Chebyshev polynomials S(n,51).
  • Sequence: 1, 52, 2651, 135149, 6889948, 351252199,
    In OEIS: - A097837 Chebyshev polynomials S(n,51) + S(n-1,51) with diophantine property.
  • Sequence: 2, 51, 2599, 132498, 6754799, 344362251,
    In OEIS: - A099368 Twice Chebyshev's polynomials of the first kind, T(n,x), evaluated at x=51/2.
• a(n) = 52a(n-1) - a(n-2).
  • Sequence: 1, 26, 1351, 70226, 3650401, 189750626,
    In OEIS: - A114052 x such that x^2 - 27*y^2 = 1.
• a(n) = 66a(n-1) - a(n-2).
  • Sequence: 1, 33, 2177, 143649, 9478657, 625447713,
    In OEIS: - A099370 Chebyshev's polynomial of the first kind, T(n,x), evaluated at x=33.
  • Sequence: 1, 65, 4289, 283009, 18674305, 1232221121,
    In OEIS: - A078988 Chebyshev sequence with diophantine property.
  • Sequence: 1, 66, 4355, 287364, 18961669, 1251182790,
    In OEIS: - A097316 Chebyshev U(n,x) polynomial evaluated at x=33.
  • Sequence: 1, 67, 4421, 291719, 19249033, 1270144459,
    In OEIS: - A078989 Chebyshev sequence with diophantine property.
  • Sequence: 0, 8, 528, 34840, 2298912, 151693352,
    In OEIS: - A121740 Solutions to the Pell equation x^2 - 17y^2 = 1 (y values).
• a(n) = 83a(n-1) - a(n-2).
  • Sequence: 1, 82, 6805, 564733, 46866034, 3889316089,
    In OEIS: - A097841 First differences of Chebyshev polynomials S(n,83)=A097839(n) with diophantine property.
  • Sequence: 1, 83, 6888, 571621, 47437655, 3936753744,
    In OEIS: - A097839 Chebyshev polynomials S(n,83).
  • Sequence: 1, 84, 6971, 578509, 48009276, 3984191399,
    In OEIS: - A097840 Chebyshev polynomials S(n,83) + S(n-1,83) with diophantine property.
  • Sequence: 2, 83, 6887, 571538, 47430767, 3936182123,
    In OEIS: - A099373 Twice Chebyshev's polynomials of the first kind, T(n,x), evaluated at 83/2.
• a(n) = 98a(n-1) - a(n-2).
  • Sequence: 1, 99, 9701, 950599, 93149001, 9127651499,
    In OEIS: - A046173 Indices of square numbers which are also pentagonal.
  • Sequence: 0, 20, 1960, 192060, 18819920, 1844160100,
    In OEIS: - A072818 Possibly the only integers of the form sqrt(m^2*(m^2-1)*2/3) [only checked for the first 5 terms].
• a(n) = 102a(n-1) - a(n-2).
  • Sequence: 1, 51, 5201, 530451, 54100801, 5517751251,
    In OEIS: - A099397 Chebyshev's polynomial of the first kind, T(n,x), evaluated at x=51.
  • Sequence: 1, 101, 10301, 1050601, 107151001, 10928351501,
    In OEIS: - A097727 Pell equation solutions (5*b(n))^2 - 26*a(n)^2 = -1 with b(n):=A097726(n), n ≥ 0.
  • Sequence: 1, 102, 10403, 1061004, 108212005, 11036563506,
    In OEIS: - A097725 Chebyshev U(n,x) polynomial evaluated at x=51.
  • Sequence: 1, 103, 10505, 1071407, 109273009, 11144775511,
    In OEIS: - A097726 Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n ≥ 0.
• a(n) = 110a(n-1) - a(n-2).
  • Sequence: 1, 55, 6049, 665335, 73180801, 8049222775,
    In OEIS: - A114049 x such that x^2 - 21*y^2 = 1.
• a(n) = 123a(n-1) - a(n-2).
  • Sequence: 1, 122, 15005, 1845493, 226980634, 27916772489,
    In OEIS: - A097843 First differences of Chebyshev polynomials S(n,123)=A049670(n+1) with diophantine property.
  • Sequence: 1, 123, 15128, 1860621, 228841255, 28145613744,
    In OEIS: - A049670 a(n)=F(10n)/55, where F=A000045 (the Fibonacci sequence).
  • Sequence: 1, 124, 15251, 1875749, 230701876, 28374454999,
    In OEIS: - A097842 Chebyshev polynomials S(n,123) + S(n-1,123) with diophantine property.
  • Sequence: 2, 123, 15127, 1860498, 228826127, 28143753123,
    In OEIS: - A065705 Lucas numbers L(10n).
• a(n) = 146a(n-1) - a(n-2).
  • Sequence: 1, 145, 21169, 3090529, 451196065, 65871534961,
    In OEIS: - A097730 Pell equation solutions (6*b(n))^2 - 37*a(n)^2 = -1 with b(n):=A097729(n), n ≥ 0.
  • Sequence: 1, 146, 21315, 3111844, 454307909, 66325842870,
    In OEIS: - A097728 Chebyshev U(n,x) polynomial evaluated at x=73 = 2*6^2+1.
  • Sequence: 1, 147, 21461, 3133159, 457419753, 66780150779,
    In OEIS: - A097729 Pell equation solutions (6*a(n))^2 - 37*b(n)^2 = -1 with b(n):=A097730(n), n ≥ 0.
• a(n) = 171a(n-1) - a(n-2).
  • Sequence: 1, 170, 29069, 4970629, 849948490, 145336221161,
    In OEIS: - A098244 First differences of Chebyshev polynomials S(n,171)=A097844(n) with diophantine property.
  • Sequence: 1, 171, 29240, 4999869, 854948359,
    In OEIS: - A097844 Chebyshev polynomials S(n,171).
  • Sequence: 1, 172, 29411, 5029109, 859948228, 147046117879,
    In OEIS: - A097845 Chebyshev polynomials S(n,171) + S(n-1,171) with diophantine property.
• a(n) = 194a(n-1) - a(n-2).
  • Sequence: 1, 195, 37829, 7338631, 1423656585,
    In OEIS: - A084232 RMS values associated with A084231.
• a(n) = 198a(n-1) - a(n-2).
  • Sequence: 1, 197, 39005, 7722793, 1529074009,
    In OEIS: - A097733 Pell equation solutions (7*b(n))^2 - 2*(5*a(n))^2 = -1 with b(n):=A097732(n), n ≥ 0. Note that D=50=2*5^2 is not square-free.
  • Sequence: 1, 198, 39203, 7761996, 1536836005,
    In OEIS: - A097731 Chebyshev U(n,x) polynomial evaluated at x=99 = 2*7^2+1.
  • Sequence: 1, 199, 39401, 7801199, 1544598001,
    In OEIS: - A097732 Pell equation solutions (7*a(n))^2 - 2*(5*b(n))^2 = -1 with b(n):=A097733(n), n ≥ 0. Note that D=50=2*5^2 is not square-free.
• a(n) = 227a(n-1) - a(n-2).
  • Sequence: 1, 226, 51301, 11645101, 2643386626,
    In OEIS: - A098247 First differences of Chebyshev polynomials S(n,227)=A098245(n) with diophantine property.
  • Sequence: 1, 227, 51528, 11696629, 2655083255,
    In OEIS: - A098245 Chebyshev polynomials S(n,227).
  • Sequence: 1, 228, 51755, 11748157, 2666779884,
    In OEIS: - A098246 Chebyshev polynomials S(n,227) + S(n-1,227) with diophantine property.
• a(n) = 258a(n-1) - a(n-2).
  • Sequence: 1, 257, 66305, 17106433, 4413393409,
    In OEIS: - A097736 Pell equation solutions (8*b(n))^2 - 65*a(n)^2 = -1 with b(n):=A097735(n), n ≥ 0.
  • Sequence: 1, 258, 66563, 17172996, 4430566405,
    In OEIS: - A097734 Chebyshev U(n,x) polynomial evaluated at x=129 = 3*43.
  • Sequence: 1, 259, 66821, 17239559, 4447739401,
    In OEIS: - A097735 Pell equation solutions (8*a(n))^2 - 65*b(n)^2 = -1 with b(n):=A097736(n), n ≥ 0.
• a(n) = 291a(n-1) - a(n-2).
  • Sequence: 1, 290, 84389, 24556909, 7145976130,
    In OEIS: - A098250 First differences of Chebyshev polynomials S(n,291)=A098248(n) with diophantine property.
  • Sequence: 1, 291, 84680, 24641589, 7170617719,
    In OEIS: - A098248 Chebyshev polynomials S(n,291).
  • Sequence: 1, 292, 84971, 24726269, 7195259308,
    In OEIS: - A098249 Chebyshev polynomials S(n,291) + S(n-1,291) with diophantine property.
• a(n) = 322a(n-1) - a(n-2).
  • Sequence: 2, 322, 103682, 33385282, 10749957122,
    In OEIS: - A089775 Lucas numbers L(12n).
• a(n) = 326a(n-1) - a(n-2).
  • Sequence: 1, 325, 105949, 34539049, 11259624025,
    In OEIS: - A097739 Pell equation solutions (9*b(n))^2 - 82*a(n)^2 = -1 with b(n):=A097738(n), n ≥ 0.
  • Sequence: 1, 326, 106275, 34645324, 11294269349,
    In OEIS: - A097737 Chebyshev U(n,x) polynomial evaluated at x=163.
  • Sequence: 1, 327, 106601, 34751599, 11328914673,
    In OEIS: - A097738 Pell equation solutions (9*a(n))^2 - 82*b(n)^2 = -1 with b(n):=A097739(n), n ≥ 0.
• a(n) = 340a(n-1) - a(n-2).
  • Sequence: 1, 170, 57799, 19651490, 6681448801,
    In OEIS: - A114048 x such that x^2 - 19*y^2 = 1.
• a(n) = 363a(n-1) - a(n-2).
  • Sequence: 1, 362, 131405, 47699653, 17314842634,
    In OEIS: - A098253 First differences of Chebyshev polynomials S(n,363)=A098251(n) with diophantine property.
  • Sequence: 1, 363, 131768, 47831421, 17362674055,
    In OEIS: - A098251 Chebyshev polynomials S(n,363).
  • Sequence: 1, 364, 132131, 47963189, 17410505476,
    In OEIS: - A098252 Chebyshev polynomials S(n,363) + S(n-1,363) with diophantine property.
• a(n) = 394a(n-1) - a(n-2).
  • Sequence: 1, 197, 77617, 30580901, 12048797377,
    In OEIS: - A114050 x such that x^2 - 22*y^2 = 1.
• a(n) = 402a(n-1) - a(n-2).
  • Sequence: 1, 401, 161201, 64802401, 26050404001,
    In OEIS: - A097742 Pell equation solutions (10*b(n))^2 - 101*a(n)^2 = -1 with b(n):=A097741(n), n ≥ 0.
  • Sequence: 1, 402, 161603, 64964004, 26115368005,
    In OEIS: - A097740 Chebyshev U(n,x) polynomial evaluated at x=201.
  • Sequence: 1, 403, 162005, 65125607, 26180332009,
    In OEIS: - A097741 Pell equation solutions (10*a(n))^2 - 101*b(n)^2 = -1 with b(n):=A097742(n), n ≥ 0.
• a(n) = 443a(n-1) - a(n-2).
  • Sequence: 1, 442, 195805, 86741173, 38426143834,
    In OEIS: - A098256 First differences of Chebyshev polynomials S(n,443)=A098254(n) with diophantine property.
  • Sequence: 1, 443, 196248, 86937421, 38513081255,
    In OEIS: - A098254 Chebyshev polynomials S(n,443).
  • Sequence: 1, 444, 196691, 87133669, 38600018676,
    In OEIS: - A098255 Chebyshev polynomials S(n,443) + S(n-1,443) with diophantine property.
• a(n) = 486a(n-1) - a(n-2).
  • Sequence: 1, 485, 235709, 114554089, 55673051545,
    In OEIS: - A097767 Pell equation solutions (11*b(n))^2 - 122*a(n)^2 = -1 with b(n):=A097766(n), n ≥ 0.
  • Sequence: 1, 486, 236195, 114790284, 55787841829,
    In OEIS: - A097765 Chebyshev U(n,x) polynomial evaluated at x=243=2*11^2+1.
  • Sequence: 1, 487, 236681, 115026479, 55902632113,
    In OEIS: - A097766 Pell equation solutions (11*a(n))^2 - 122*b(n)^2 = -1 with b(n):=A097767(n), n ≥ 0.
• a(n) = 531a(n-1) - a(n-2).
  • Sequence: 1, 530, 281429, 149438269, 79351439410,
    In OEIS: - A098259 First differences of Chebyshev polynomials S(n,531)=A098257(n) with diophantine property.
  • Sequence: 1, 531, 281960, 149720229, 79501159639,
    In OEIS: - A098257 Chebyshev polynomials S(n,531).
  • Sequence: 1, 532, 282491, 150002189, 79650879868,
    In OEIS: - A098258 Chebyshev polynomials S(n,531) + S(n-1,531) with diophantine property.
• a(n) = 578a(n-1) - a(n-2).
  • Sequence: 1, 577, 333505, 192765313, 111418017409,
    In OEIS: - A097770 Pell equation solutions (12*b(n))^2 - 145*a(n)^2 = -1 with b(n):=A097769(n), n ≥ 0.
  • Sequence: 1, 578, 334083, 193099396, 111611116805,
    In OEIS: - A097768 Chebyshev U(n,x) polynomial evaluated at x=289=2*12^2+1.
  • Sequence: 1, 579, 334661, 193433479, 111804216201,
    In OEIS: - A097769 Pell equation solutions (12*a(n))^2 - 145*b(n)^2 = -1 with b(n):=A097770(n), n ≥ 0.
• a(n) = 627a(n-1) - a(n-2).
  • Sequence: 1, 626, 392501, 246097501, 154302740626,
    In OEIS: - A098262 First differences of Chebyshev polynomials S(n,627)=A098260(n) with diophantine property.
  • Sequence: 1, 627, 393128, 246490629, 154549231255,
    In OEIS: - A098260 Chebyshev polynomials S(n,627).
  • Sequence: 1, 628, 393755, 246883757, 154795721884,
    In OEIS: - A098261 Chebyshev polynomials S(n,627) + S(n-1,627) with diophantine property.
• a(n) = 678a(n-1) - a(n-2).
  • Sequence: 1, 677, 459005, 311204713, 210996336409,
    In OEIS: - A097773 Pell equation solutions (13*b(n))^2 - 170*a(n)^2 = -1 with b(n):=A097772(n), n ≥ 0.
  • Sequence: 1, 678, 459683, 311664396, 211308000805,
    In OEIS: - A097771 Chebyshev U(n,x) polynomial evaluated at x=339=2*13^2+1.
  • Sequence: 1, 679, 460361, 312124079, 211619665201,
    In OEIS: - A097772 Pell equation solutions (13*a(n))^2 - 170*b(n)^2 = -1 with b(n):=A097771(n), n ≥ 0.
• a(n) = 731a(n-1) - a(n-2).
  • Sequence: 1, 730, 533629, 390082069, 285149458810,
    In OEIS: - A098292 First differences of Chebyshev polynomials S(n,731)=A098263(n) with diophantine property.
  • Sequence: 1, 731, 534360, 390616429, 285540075239,
    In OEIS: - A098263 Chebyshev polynomials S(n,731).
  • Sequence: 1, 732, 535091, 391150789, 285930691668,
    In OEIS: - A098291 Chebyshev polynomials S(n,731) + S(n-1,731) with diophantine property.
• a(n) = 786a(n-1) - a(n-2).
  • Sequence: 1, 785, 617009, 484968289, 381184458145,
    In OEIS: - A097776 Pell equation solutions (14*b(n))^2 - 197*a(n)^2 = -1 with b(n):=A097775(n), n ≥ 0.
  • Sequence: 1, 786, 617795, 485586084, 381670044229,
    In OEIS: - A097774 Chebyshev U(n,x) polynomial evaluated at x=393=2*14^2+1.
  • Sequence: 1, 787, 618581, 486203879, 382155630313,
    In OEIS: - A097775 Pell equation solutions (14*a(n))^2 - 197*b(n)^2 = -1 with b(n):=A097776(n), n ≥ 0.
• a(n) = 1298a(n-1) - a(n-2).
  • Sequence: 0, 180, 233640, 303264540, 393637139280,
    In OEIS: - A075871 Numbers n such that 13*n^2 + 1 is a square.
  • Sequence: 1, 649, 842401, 1093435849, 1419278889601,
    In OEIS: - A114047 x such that x^2 - 13*y^2 = 1.
• a(n) = 2302a(n-1) - a(n-2).
  • Sequence: 1, 1151, 2649601, 6099380351, 14040770918401,
    In OEIS: - A114046 x such that x^2 - 92*y^2 = 1.
• a(n) = 2702a(n-1) - a(n-2).
  • Sequence: -15, 15, 40545, 109552575, 296011017105,
    In OEIS: - A094836 a(n)=2702*a(n-1) - a(n-2); a(-1)=-15; a(0)=15.
  • Sequence: 26, 26, 70226, 189750626, 512706121226,
    In OEIS: - A094835 a(n) = 2702*a(n-1) - a(n-2); a(-1) = a(0) = 26.

a(n) = d * a(n-1) + d * a(n-2).

Common properties for sequences with initial terms a(0) = 1 and a(1) = d+1 :

• a(n) equals the number of 00-avoiding words of length n on alphabet {0,1,2, ..., d}. Proof.
• The generating function of this sequence is (1+x)/(1-dx-dx²).

Common properties for sequences with initial terms a(0) = 0, a(1) = 1:

• The generating function of this sequence is x/(1-dx-dx²).
• a(n) = (pⁿ - qⁿ)/(p-q), where p and q are the roots of the equation: x² - dx - d = 0.

Common properties for sequences with initial terms a(0) = 2, a(1) = d:

• The generating function of this sequence is (2-dx)/(1-dx-dx²).
• a(n) = pⁿ + qⁿ, where p and q are the roots of the equation: x² - dx - d = 0. Namely a(n) = = ((d+sqrt(d²+4d))/2)ⁿ + ((d-sqrt(d²+4d))/2)ⁿ. Proof.

Sequences:

• a(n) = - 4a(n-1) - 4a(n-2).
  
  • Sequence: 1, -2, 4, -8, 16, -32, 64, -128, 256,
    In OEIS: - A122803 Powers of -2.
  • Sequence: 0, -1, 4, -12, 32, -80, 192, -448, 1024,
    In OEIS: - A085750 Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 ≤ i,j ≤ n.
• a(n) = - 2a(n-1) - 2a(n-2).
  Sequence: 1, -3, 4, -2, -4, 12, -16, 8, 16, -48,
  In OEIS: - A078069 Expansion of (1-x)/(1+2*x+2*x^2).
• a(n) = - a(n-1) - a(n-2).
  Sequence: 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1,
  In OEIS: - A061347 Period 3.
• a(n) = a(n-1) + a(n-2). For d = 1 see a(n) = a(n-1) + a(n-2).
• a(n) = 2a(n-1) + 2a(n-2).
  
  • Sequence: 1, 1, 4, 10, 28, 76, 208, 568, 1552, 4240,
    In OEIS: - A026150 a(0) = a(1) = 1; a(n+2) = 2*a(n+1) + 2*a(n).
  • Sequence: 1, 2, 6, 16, 44, 120, 328, 896, 2448,
    In OEIS: - A002605 a(n+2) = 2*a(n+1) + 2*a(n). Also A080953 a(n)=2(a(n-1)+a(n-2)), a(0)=0, a(1)=1.
  • Sequence: 1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136,
    In OEIS: - A028859 a(n+2) = 2a(n+1) + 2a(n).
  • Sequence: 2, 2, 8, 20, 56, 152, 416, 1136, 3104, 8480,
    In OEIS: - A080040 a(n)=2a(n-1)+2a(n-2), a(0)=2, a(1)=2.
  • Sequence: 0, 2, 4, 12, 32, 88, 240, 656, 1792, 4896,
    In OEIS: - A106433 Yet another way to compute A028860 ( first two terms different) : 2 X 2 vector Matrix Markov with characteristic Polynomial: x^2-2*x-2.
  • Sequence: 0, 3, 6, 18, 48, 132, 360, 984, 2688, 7344,
    In OEIS: - A083337 a(n)=2a(n-1)+2a(n-2).
  • Sequence: 4, 11, 30, 82, 224, 612, 1672, 4568, 12480,
    In OEIS: - A021006 Pisot sequence P(4,11), a(0)=4, a(1)=11, a(n+1) is the nearest integer to a(n)^2/a(n-1). Evidently satisfies a(n)=2a(n-1)+2a(n-2).
  • Sequence: 0, 4, 8, 24, 64, 176, 480, 1312, 3584,
    In OEIS: - A116556 a(0)=0, a(1)=4, a(n)=2a(n-1)+2a(n-2)
• a(n) = 3a(n-1) + 3a(n-2).
  
  • Sequence: 1, 3, 12, 45, 171, 648, 2457,
    In OEIS: - A030195 a(n) = 3*a(n-1)+3*a(n-2), a(0)=0, a(1)=1.
  • Sequence: 1, 4, 15, 57, 216, 819,
    In OEIS: - A125145 a(n) = 3a(n-1) + 3a(n-2). a(0) = 1, a(1) = 4.
  • Sequence: 0, 3, 9, 36, 135, 513, 1944,
    In OEIS: - A106435 a(n) = 3*a(n-1)+3*a(n-2), a(0)=0, a(1)=3.
  • Sequence: 3, 15, 54, 207, 783, 2970, 11259,
    In OEIS: - A085480 a(n) = p^n + q^n, where p = (3 + sqrt 21)/2, q = (3 - sqrt 21)/2.
  • Sequence: 1, 6, 21, 81, 306, 1161, 4401, 16686,
    In OEIS: - A108306 Expansion of (3*x+1)/(1-3*x-3*x^2).
• a(n) = 4a(n-1) + 4a(n-2).
  
  • Sequence: 1, 4, 20, 96, 464, 2240, 10816, 52224,
    In OEIS: - A057087 Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence.
  • Sequence: 1, 5, 24, 116, 560, 2704, 13056, 63040,
    In OEIS: - A086347 On a 3 X 3 board, number of n-move routes of chess king ending at a given side cell.
  • Sequence: 1, 2, 12, 56, 272, 1312, 6336, 30592,
    In OEIS: - A084128 Generalized Fibonacci sequence.
  • Sequence: 1, 0, 4, 16, 80, 384, 1856, 8960, 43264,
    In OEIS: - A094013 Expansion of (1-4x)/(1-4x-4x^2). Also A106568 First entry of the vector (M^n)v, where M is the 2x2 matrix [[0,4],[1,4]] and v is the column vector [0,1].
• a(n) = 5a(n-1) + 5a(n-2).
  
  • Sequence: 1, 5, 30, 175, 1025, 6000, 35125, 205625,
    In OEIS: - A057088 Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence.
  • Sequence: 0, 5, 25, 150, 875, 5125, 30000, 175625,
    In OEIS: - A106565 Let M={{0, 5}, {1, 5}}, v[n]=M.v[n-1]; then a(n) =v[n][[1]].
• a(n) = 6a(n-1) + 6a(n-2).
  
  • Sequence: 1, 6, 42, 288, 1980, 13608, 93528, 642816,
    In OEIS: - A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.
  • Sequence: 8, 55, 378, 2598, 17856, 122724, 843480,
    In OEIS: - A010924 Pisot sequence E(8,55), a(n)=[ a(n-1)^2/a(n-2)+1/2 ].
• a(n) = 7a(n-1) + 7a(n-2).
  Sequence: 1, 7, 56, 441, 3479, 27440, 216433,
  In OEIS: - A057090 Scaled Chebyshev U-polynomials evaluated at i*sqrt(7)/2. Generalized Fibonacci sequence.
• a(n) = 8a(n-1) + 8a(n-2).
  Sequence: 1, 8, 72, 640, 5696, 50688, 451072,
  In OEIS: - A057091 Scaled Chebyshev U-polynomials evaluated at i*sqrt(2). Generalized Fibonacci sequence.
• a(n) = 9a(n-1) + 9a(n-2).
  Sequence: 1, 9, 90, 891, 8829, 87480, 866781,
  In OEIS: - A057092 Scaled Chebyshev U-polynomials evaluated at i*3/2. Generalized Fibonacci sequence.
• a(n) = 10a(n-1) + 10a(n-2).
  Sequence: 1, 10, 110, 1200, 13100, 143000, 1561000,
  In OEIS: - A057093 Scaled Chebyshev U-polynomials evaluated at i*sqrt(10)/2. Generalized Fibonacci sequence.

a(n) = a(n-1) + a(n-2).

Common properties for these sequences:

• The sum of the first n terms: a(1) + a(2) + ... + a(n-1) + a(n) equals a(n+2) - a(2).
• The sum of the first n odd numbered terms: a(1) + a(3) + ... + a(2n-3) + a(2n-1) equals a(2n) - a(2) + a(1).
• The sum of the first n even numbered terms: a(2) + a(4) + ... + a(2n-2) + a(2n) equals a(2n+1) - a(1).

Properties for the sequence with initial terms a(0) = 1 and a(1) = 2 (shifted Fibonacci sequence):

• a(n) equals the number of 00-avoiding words of length n on alphabet {0,1}.
• The generating function of this sequence is (1+x)/(1-x-x²).

Properties for the sequence with initial terms a(0) = 0, a(1) = 1 (Fibonacci sequence):

• The generating function of this sequence is x/(1-x-x²).
• a(n) = (pⁿ - qⁿ)/(p-q), where p and q are the roots of the equation: x² - x - 1 = 0. Namely a(n) = (φⁿ + (-1/φ)ⁿ)/sqrt(5), where φ is the golden ratio.
• Asymptotically a(n) = Round(φⁿ/sqrt(5)), where φ is the golden ratio.

Properties for the sequence with initial terms a(0) = 2, a(1) = 1 (Lucas sequence):

• The generating function of this sequence is (2-x)/(1-x-x²).
• a(n) = pⁿ + qⁿ, where p and q are the roots of the equation: x² - x - 1 = 0. Namely a(n) = ((1+sqrt(5))/2)ⁿ + ((1-sqrt(5))/2)ⁿ = φⁿ + (-1/φ)ⁿ, where φ is the golden ratio.
• Asymptotically a(n) = Round(φⁿ), where φ is the golden ratio.

Sequences:

• a(n) = a(n-1) + a(n-2).
  • Sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
    In OEIS: - A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2), F(0) = 0, F(1) = 1. Also A020695 Pisot sequence E(2,3). Also A020701 Pisot sequences E(3,5), P(3,5). Also A020712 Pisot sequences E(5,8), P(5,8).
  • Sequence: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123,
    In OEIS: - A000032 Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2); and A000204 Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.
  • Sequence: 1, 4, 5, 9, 14, 23, 37, 60, 97, 157,
    In OEIS: - A000285 a(n) = a(n-1) + a(n-2). Also A104449 Fibonacci-type sequence. Each term is the sum of the two previous terms.
  • Sequence: 1, 5, 6, 11, 17, 28, 45, 73, 118, 191,
    In OEIS: - A022095 Fibonacci sequence beginning 1 5.
  • Sequence: 2, 4, 6, 10, 16, 26, 42, 68, 110, 178,
    In OEIS: - A090991 Number of meaningful differential operations of the n-th order on the space R^6. Also A118658 L_n - F_n where L_n is the Lucas Number and F_n is the Fibonacci Number. Also A078642 Numbers with two representations as the sum of two Fibonacci numbers.
  • Sequence: 2, 5, 7, 12, 19, 31, 50, 81, 131, 212,
    In OEIS: - A001060 a(n) = a(n-1) + a(n-2); and A013655 F(n)+L(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.
  • Sequence: 2, 6, 8, 14, 22, 36, 58, 94, 152, 246,
    In OEIS: - A022112 Fibonacci sequence beginning 2 6.
  • Sequence: 2, 7, 9, 16, 25, 41, 66, 107, 173, 280,
    In OEIS: - A022113 Fibonacci sequence beginning 2 7.
  • Sequence: 2, 9, 11, 20, 31, 51, 82, 133, 215,
    In OEIS: - A022114 Fibonacci sequence beginning 2 9.
  • Sequence: 2, 10, 12, 22, 34, 56, 90, 146, 236,
    In OEIS: - A022367 Fibonacci sequence beginning 2 10.
  • Sequence: 2, 11, 13, 24, 37, 61, 98, 159, 257,
    In OEIS: - A022115 Fibonacci sequence beginning 2 11.
  • Sequence: 2, 12, 14, 26, 40, 66, 106, 172, 278,
    In OEIS: - A022368 Fibonacci sequence beginning 2 12.
  • Sequence: 2, 13, 15, 28, 43, 71, 114, 185, 299,
    In OEIS: - A022116 Fibonacci sequence beginning 2 13.
  • Sequence: 2, 14, 16, 30, 46, 76, 122, 198, 320,
    In OEIS: - A022369 Fibonacci sequence beginning 2 14.
  • Sequence: 2, 15, 17, 32, 49, 81, 130, 211, 341,
    In OEIS: - A022117 Fibonacci sequence beginning 2 15.
  • Sequence: 2, 16, 18, 34, 52, 86, 138, 224, 362,
    In OEIS: - A022370 Fibonacci sequence beginning 2 16.
  • Sequence: 2, 17, 19, 36, 55, 91, 146, 237, 383,
    In OEIS: - A022118 Fibonacci sequence beginning 2 17.
  • Sequence: 2, 18, 20, 38, 58, 96, 154, 250, 404,
    In OEIS: - A022371 Fibonacci sequence beginning 2 18.
  • Sequence: 2, 19, 21, 40, 61, 101, 162, 263, 425,
    In OEIS: - A022119 Fibonacci sequence beginning 2 19.
  • Sequence: 2, 20, 22, 42, 64, 106, 170, 276, 446,
    In OEIS: - A022372 Fibonacci sequence beginning 2 20.
  • Sequence: 2, 22, 24, 46, 70, 116, 186, 302, 488,
    In OEIS: - A022373 Fibonacci sequence beginning 2 22.
  • Sequence: 2, 24, 26, 50, 76, 126, 202, 328, 530,
    In OEIS: - A022374 Fibonacci sequence beginning 2 24.
  • Sequence: 2, 26, 28, 54, 82, 136, 218, 354, 572,
    In OEIS: - A022375 Fibonacci sequence beginning 2 26.
  • Sequence: 2, 28, 30, 58, 88, 146, 234, 380, 614,
    In OEIS: - A022376 Fibonacci sequence beginning 2 28.
  • Sequence: 2, 30, 32, 62, 94, 156, 250, 406, 656,
    In OEIS: - A022377 Fibonacci sequence beginning 2 30.
  • Sequence: 2, 32, 34, 66, 100, 166, 266, 432, 698,
    In OEIS: - A022378 Fibonacci sequence beginning 2 32.
  • Sequence: 0, 3, 3, 6, 9, 15, 24, 39, 63, 102,
    In OEIS: - A022086 Fibonacci sequence beginning 0 3.
  • Sequence: 3, 7, 10, 17, 27, 44, 71, 115, 186, 301,
    In OEIS: - A022120 Fibonacci sequence beginning 3 7.
  • Sequence: 3, 8, 11, 19, 30, 49, 79, 128, 207, 335,
    In OEIS: - A022121 Fibonacci sequence beginning 3 8.
  • Sequence: 3, 9, 12, 21, 33, 54, 87, 141, 228, 369,
    In OEIS: - A022379 Fibonacci sequence beginning 3 9.
  • Sequence: 3, 10, 13, 23, 36, 59, 95, 154, 249, 403,
    In OEIS: - A022122 Fibonacci sequence beginning 3 10.
  • Sequence: 3, 11, 14, 25, 39, 64, 103, 167, 270, 437,
    In OEIS: - A022123 Fibonacci sequence beginning 3 11.
  • Sequence: 3, 12, 15, 27, 42, 69, 111, 180, 291, 471,
    In OEIS: - A022380 Fibonacci sequence beginning 3 12.
  • Sequence: 3, 13, 16, 29, 45, 74, 119, 193, 312, 505,
    In OEIS: - A022124 Fibonacci sequence beginning 3 13.
  • Sequence: 3, 14, 17, 31, 48, 79, 127, 206, 333, 539,
    In OEIS: - A022125 Fibonacci sequence beginning 3 14.
  • Sequence: 3, 15, 18, 33, 51, 84, 135, 219, 354, 573,
    In OEIS: - A022381 Fibonacci sequence beginning 3 15.
  • Sequence: 3, 16, 19, 35, 54, 89, 143, 232, 375, 607,
    In OEIS: - A022126 Fibonacci sequence beginning 3 16.
  • Sequence: 3, 17, 20, 37, 57, 94, 151, 245, 396, 641,
    In OEIS: - A022127 Fibonacci sequence beginning 3 17.
  • Sequence: 3, 19, 22, 41, 63, 104, 167, 271, 438, 709,
    In OEIS: - A022128 Fibonacci sequence beginning 3 19.
  • Sequence: 3, 20, 23, 43, 66, 109, 175, 284, 459, 743,
    In OEIS: - A022129 Fibonacci sequence beginning 3 20.
  • Sequence: 0, 4, 4, 8, 12, 20, 32, 52, 84, 136, 220,
    In OEIS: - A022087 Fibonacci sequence beginning 0 4.
  • Sequence: 4, 9, 13, 22, 35, 57, 92, 149, 241, 390,
    In OEIS: - A022130 Fibonacci sequence beginning 4 9.
  • Sequence: 4, 10, 14, 24, 38, 62, 100, 162, 262, 424,
    In OEIS: - A022382 Fibonacci sequence beginning 4 10.
  • Sequence: 4, 11, 15, 26, 41, 67, 108, 175, 283, 458,
    In OEIS: - A022131 Fibonacci sequence beginning 4 11.
  • Sequence: 4, 13, 17, 30, 47, 77, 124, 201, 325, 526,
    In OEIS: - A022132 Fibonacci sequence beginning 4 13.
  • Sequence: 4, 14, 18, 32, 50, 82, 132, 214, 346, 560,
    In OEIS: - A022383 Fibonacci sequence beginning 4 14.
  • Sequence: 4, 15, 19, 34, 53, 87, 140, 227, 367, 594,
    In OEIS: - A022133 Fibonacci sequence beginning 4 15.
  • Sequence: 4, 17, 21, 38, 59, 97, 156, 253, 409, 662,
    In OEIS: - A022134 Fibonacci sequence beginning 4 17.
  • Sequence: 4, 18, 22, 40, 62, 102, 164, 266, 430, 696,
    In OEIS: - A022384 Fibonacci sequence beginning 4 18.
  • Sequence: 4, 19, 23, 42, 65, 107, 172, 279, 451, 730,
    In OEIS: - A022135 Fibonacci sequence beginning 4 19.
  • Sequence: 4, 22, 26, 48, 74, 122, 196, 318, 514, 832,
    In OEIS: - A022385 Fibonacci sequence beginning 4 22.
  • Sequence: 4, 26, 30, 56, 86, 142, 228, 370, 598, 968,
    In OEIS: - A022386 Fibonacci sequence beginning 4 26.
  • Sequence: 4, 30, 34, 64, 98, 162, 260, 422, 682, 1104,
    In OEIS: - A022387 Fibonacci sequence beginning 4 30.
  • Sequence: 0, 5, 5, 10, 15, 25, 40, 65, 105, 170, 275,
    In OEIS: - A022088 Fibonacci sequence beginning 0 5.
  • Sequence: 5, 11, 16, 27, 43, 70, 113, 183, 296, 479,
    In OEIS: - A022136 Fibonacci sequence beginning 5 11.
  • Sequence: 5, 12, 17, 29, 46, 75, 121, 196, 317, 513,
    In OEIS: - A022137 Fibonacci sequence beginning 5 12.
  • Sequence: 5, 13, 18, 31, 49, 80, 129, 209, 338, 547,
    In OEIS: - A022138 Fibonacci sequence beginning 5 13.
  • Sequence: 5, 14, 19, 33, 52, 85, 137, 222, 359, 581,
    In OEIS: - A022139 Fibonacci sequence beginning 5 14.
  • Sequence: 5, 16, 21, 37, 58, 95, 153, 248, 401, 649,
    In OEIS: - A022140 Fibonacci sequence beginning 5 16.
  • Sequence: 5, 17, 22, 39, 61, 100, 161, 261, 422, 683,
    In OEIS: - A022141 Fibonacci sequence beginning 5 17.
  • Sequence: 5, 18, 23, 41, 64, 105, 169, 274, 443, 717,
    In OEIS: - A022142 Fibonacci sequence beginning 5 18.
  • Sequence: 5, 19, 24, 43, 67, 110, 177, 287, 464, 751,
    In OEIS: - A022143 Fibonacci sequence beginning 5 19.
  • Sequence: 5, 19, 24, 43, 67, 110, 177, 287, 464, 751,
    In OEIS: - A022143 Fibonacci sequence beginning 5 19.
  • Sequence: 0, 6, 6, 12, 18, 30, 48, 78, 126, 204, 330,
    In OEIS: - A022089 Fibonacci sequence beginning 0 6.
  • Sequence: 1, 6, 7, 13, 20, 33, 53, 86, 139, 225, 364,
    In OEIS: - A022096 Fibonacci sequence beginning 1 6.
  • Sequence: 6, 13, 19, 32, 51, 83, 134, 217, 351, 568,
    In OEIS: - A022388 Fibonacci sequence beginning 6 13.
  • Sequence: 0, 7, 7, 14, 21, 35, 56, 91, 147, 238, 385,
    In OEIS: - A022090 Fibonacci sequence beginning 0 7.
  • Sequence: 1, 7, 8, 15, 23, 38, 61, 99, 160, 259, 419,
    In OEIS: - A022097 Fibonacci sequence beginning 1 7.
  • Sequence: 7, 15, 22, 37, 59, 96, 155, 251, 406, 657,
    In OEIS: - A022389 Fibonacci sequence beginning 7 15.
  • Sequence: 7, 26, 33, 59, 92, 151, 243, 394, 637, 1031,
    In OEIS: - A098127 Fibonacci sequence with a(1)=7 and a(2) = 26.
  • Sequence: 0, 8, 8, 16, 24, 40, 64, 104, 168, 272, 440,
    In OEIS: - A022091 Fibonacci sequence beginning 0 8.
  • Sequence: 1, 8, 9, 17, 26, 43, 69, 112, 181, 293, 474,
    In OEIS: - A022098 Fibonacci sequence beginning 1 8.
  • Sequence: 8, 17, 25, 42, 67, 109, 176, 285, 461, 746,
    In OEIS: - A022390 Fibonacci sequence beginning 8 17.
  • Sequence: 0, 9, 9, 18, 27, 45, 72, 117, 189, 306, 495,
    In OEIS: - A022092 Fibonacci sequence beginning 0 9.
  • Sequence: 1, 9, 10, 19, 29, 48, 77, 125, 202, 327, 529,
    In OEIS: - A022099 Fibonacci sequence beginning 1 9.
  • Sequence: 0, 10, 10, 20, 30, 50, 80, 130, 210, 340,
    In OEIS: - A022093 Fibonacci sequence beginning 0 10.
  • Sequence: 1, 10, 11, 21, 32, 53, 85, 138, 223, 361,
    In OEIS: - A022100 Fibonacci sequence beginning 1 10.
  • Sequence: 0, 11, 11, 22, 33, 55, 88, 143, 231, 374,
    In OEIS: - A022345 Fibonacci sequence beginning 0 11.
  • Sequence: 1, 11, 12, 23, 35, 58, 93, 151, 244, 395,
    In OEIS: - A022101 Fibonacci sequence beginning 1 11.
  • Sequence: 11, 23, 34, 57, 91, 148, 239, 387, 626, 1013,
    In OEIS: - A097657 Fibonacci sequence with first two terms 11 and 23.
  • Sequence: 0, 12, 12, 24, 36, 60, 96, 156, 252, 408,
    In OEIS: - A022346 Fibonacci sequence beginning 0 12.
  • Sequence: 1, 12, 13, 25, 38, 63, 101, 164, 265, 429,
    In OEIS: - A022102 Fibonacci sequence beginning 1 12.
  • Sequence: 12, 67, 79, 146, 225, 371, 596, 967, 1563,
    In OEIS: - A091074 Fibonacci-like sequence beginning (12, 67).
  • Sequence: 0, 13, 13, 26, 39, 65, 104, 169, 273, 442,
    In OEIS: - A022347 Fibonacci sequence beginning 0 13.
  • Sequence: 1, 13, 14, 27, 41, 68, 109, 177, 286, 463,
    In OEIS: - A022103 Fibonacci sequence beginning 1 13.
  • Sequence: 0, 14, 14, 28, 42, 70, 112, 182, 294, 476,
    In OEIS: - A022348 Fibonacci sequence beginning 0 14.
  • Sequence: 1, 14, 15, 29, 44, 73, 117, 190, 307, 497,
    In OEIS: - A022104 Fibonacci sequence beginning 1 14.
  • Sequence: 0, 15, 15, 30, 45, 75, 120, 195, 315, 510,
    In OEIS: - A022349 Fibonacci sequence beginning 0 15.
  • Sequence: 1, 15, 16, 31, 47, 78, 125, 203, 328, 531,
    In OEIS: - A022105 Fibonacci sequence beginning 1 15.
  • Sequence: 0, 16, 16, 32, 48, 80, 128, 208, 336, 544,
    In OEIS: - A022350 Fibonacci sequence beginning 0 16.
  • Sequence: 1, 16, 17, 33, 50, 83, 133, 216, 349, 565,
    In OEIS: - A022106 Fibonacci sequence beginning 1 16.
  • Sequence: 0, 17, 17, 34, 51, 85, 136, 221, 357, 578,
    In OEIS: - A022351 Fibonacci sequence beginning 0 17.
  • Sequence: 1, 17, 18, 35, 53, 88, 141, 229, 370, 599,
    In OEIS: - A022107 Fibonacci sequence beginning 1 17.
  • Sequence: 0, 18, 18, 36, 54, 90, 144, 234, 378, 612,
    In OEIS: - A022352 Fibonacci sequence beginning 0 18.
  • Sequence: 1, 18, 19, 37, 56, 93, 149, 242, 391, 633,
    In OEIS: - A022108 Fibonacci sequence beginning 1 18.
  • Sequence: 0, 19, 19, 38, 57, 95, 152, 247, 399, 646,
    In OEIS: - A022353 Fibonacci sequence beginning 0 19.
  • Sequence: 1, 19, 20, 39, 59, 98, 157, 255, 412, 667,
    In OEIS: - A022109 Fibonacci sequence beginning 1 19.
  • Sequence: 0, 20, 20, 40, 60, 100, 160, 260, 420, 680,
    In OEIS: - A022354 Fibonacci sequence beginning 0 20.
  • Sequence: 1, 20, 21, 41, 62, 103, 165, 268, 433, 701,
    In OEIS: - A022110 Fibonacci sequence beginning 1 20.
  • Sequence: 0, 21, 21, 42, 63, 105, 168, 273, 441, 714,
    In OEIS: - A022355 Fibonacci sequence beginning 0 21.
  • Sequence: 1, 21, 22, 43, 65, 108, 173, 281, 454, 735,
    In OEIS: - A022391 Fibonacci sequence beginning 1 21.
  • Sequence: 0, 22, 22, 44, 66, 110, 176, 286, 462, 748,
    In OEIS: - A022356 Fibonacci sequence beginning 0 22.
  • Sequence: 1, 22, 23, 45, 68, 113, 181, 294, 475, 769,
    In OEIS: - A022392 Fibonacci sequence beginning 1 22.
  • Sequence: 0, 23, 23, 46, 69, 115, 184, 299, 483, 782,
    In OEIS: - A022357 Fibonacci sequence beginning 0 23.
  • Sequence: 1, 23, 24, 47, 71, 118, 189, 307, 496, 803,
    In OEIS: - A022393 Fibonacci sequence beginning 1 23.
  • Sequence: 0, 24, 24, 48, 72, 120, 192, 312, 504, 816,
    In OEIS: - A022358 Fibonacci sequence beginning 0 24.
  • Sequence: 1, 24, 25, 49, 74, 123, 197, 320, 517, 837,
    In OEIS: - A022394 Fibonacci sequence beginning 1 24.
  • Sequence: 0, 25, 25, 50, 75, 125, 200, 325, 525, 850,
    In OEIS: - A022359 Fibonacci sequence beginning 0 25.
  • Sequence: 1, 25, 26, 51, 77, 128, 205, 333, 538, 871,
    In OEIS: - A022395 Fibonacci sequence beginning 1 25.
  • Sequence: 0, 26, 26, 52, 78, 130, 208, 338, 546, 884,
    In OEIS: - A022360 Fibonacci sequence beginning 0 26.
  • Sequence: 1, 26, 27, 53, 80, 133, 213, 346, 559, 905,
    In OEIS: - A022396 Fibonacci sequence beginning 1 26.
  • Sequence: 0, 27, 27, 54, 81, 135, 216, 351, 567, 918,
    In OEIS: - A022361 Fibonacci sequence beginning 0 27.
  • Sequence: 1, 27, 28, 55, 83, 138, 221, 359, 580, 939,
    In OEIS: - A022397 Fibonacci sequence beginning 1 27.
  • Sequence: 0, 28, 28, 56, 84, 140, 224, 364, 588, 952,
    In OEIS: - A022362 Fibonacci sequence beginning 0 28.
  • Sequence: 1, 28, 29, 57, 86, 143, 229, 372, 601, 973,
    In OEIS: - A022398 Fibonacci sequence beginning 1 28.
  • Sequence: 0, 29, 29, 58, 87, 145, 232, 377, 609, 986,
    In OEIS: - A022363 Fibonacci sequence beginning 0 29.
  • Sequence: 1, 29, 30, 59, 89, 148, 237, 385, 622,
    In OEIS: - A022399 Fibonacci sequence beginning 1 29.
  • Sequence: 0, 30, 30, 60, 90, 150, 240, 390, 630, 1020,
    In OEIS: - A022364 Fibonacci sequence beginning 0 30.
  • Sequence: 1, 30, 31, 61, 92, 153, 245, 398, 643, 1041,
    In OEIS: - A022400 Fibonacci sequence beginning 1 30.
  • Sequence: 0, 31, 31, 62, 93, 155, 248, 403, 651, 1054,
    In OEIS: - A022365 Fibonacci sequence beginning 0 31.
  • Sequence: 1, 31, 32, 63, 95, 158, 253, 411, 664, 1075,
    In OEIS: - A022401 Fibonacci sequence beginning 1 31.
  • Sequence: 0, 32, 32, 64, 96, 160, 256, 416, 672, 1088,
    In OEIS: - A022366 Fibonacci sequence beginning 0 32.
  • Sequence: 1, 32, 33, 65, 98, 163, 261, 424, 685, 1109,
    In OEIS: - A022402 Fibonacci sequence beginning 1 32.
  • Sequence: 110, 211, 321, 532, 853, 1385, 2238, 3623,
    In OEIS: - A120727 a(n) = a(n-1) + a(n-2), starting with 110, 211.
  • Sequence: 407389224418, 76343678551, 483732902969,
    In OEIS: - A082411 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
  • Sequence: 20615674205555510, 3794765361567513, 24410439567123023,
    In OEIS: - A083216 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
  • Sequence: 62638280004239857, 49463435743205655, 112101715747445512,
    In OEIS: - A083105 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
  • Sequence: 331635635998274737472200656430763, 1510028911088401971189590305498785,
    In OEIS: - A083104 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
  • Sequence: 31786772701928802632268715130455793, 1059683225053915111058165141686995,
    In OEIS: - A083103 Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).

a(n) = d * a(n-1) + a(n-2).

Common properties for sequences starting 1, d+1:

• The generating function of this sequence is (1+x)/(1-dx-x²).
• a(n) is the number of words in the alphabet {0, 1, 2, ..., d} avoiding "11", "22", ..., "dd". Proof.

Common properties for sequences starting 2, d:

• The generating function of this sequence is (2-d)/(1-dx-x²).
• a(n) = pⁿ + qⁿ, where p and q are the roots of the equation: x² - dx - 1 = 0. Namely a(n) = ((d+sqrt(d²+4))/2)ⁿ + ((d-sqrt(d²+4))/2)ⁿ. Proof.
• Asymptotically a(n) = Round(pⁿ), where p is the largest root of the equation: x² - dx - 1 = 0. That is a(n) approaches ((d+sqrt(d²+4))/2)ⁿ.

Common properties for sequences with positive d = 2k.

• Sequence starting with 1 and k:
  • The generating function of this sequence is (1-k)/(1-2kx-x²).
  • a(n) = (pⁿ + qⁿ)/2, where p and q are the roots of the equation: x² - 2kx - 1 = 0. (Namely p = k + sqrt(k²+1) and q = k - sqrt(k²+1)).
  • a(n) are numerators of continued fractions converging to the square root of k²+1. Proof.
• Sequence starting with 1 and 2k:
  • The generating function of this sequence is 1/(1-2kx-x²).
  • a(n) are denominators of continued fractions converging to the square root of k²+1. Proof.

Sequences:

• a(n) = -11a(n-1) + a(n-2).
  • Sequence: 1, 1, -10, 111, -1231, 13652, -151403,
    In OEIS: - A122574 a(1)=a(2)=1, a(n)=-11a(n-1)+a(n-2)
• a(n) = -4a(n-1) + a(n-2).
  • Sequence: 1, -5, 21, -89, 377, -1597, 6765, -28657,
    In OEIS: - A099843 A transform of the Fibonacci numbers.
• a(n) = -2a(n-1) + a(n-2).
  • Sequence: 1, -2, 5, -12, 29, -70, 169, -408, 985,
    In OEIS: - A077985 Expansion of 1/(1+2*x-x^2).
  • Sequence: 1, -3, 4, -7, 11, -18, 29, -47, 76, -123,
    In OEIS: - A075193 "Inverted" Lucas numbers (see Comments).
• a(n) = -a(n-1) + a(n-2).
  • Sequence: 1, 2, -1, 3, -4, 7, -11, 18, -29, 47,
    In OEIS: - A061084 Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).
• a(n) = a(n-2). For d = 0 see TBD.
• a(n) = a(n-1) + a(n-2). For d = 1 see a(n) = a(n-1) + a(n-2).
• a(n) = 2a(n-1) + a(n-2).
  • Sequence: 1, 2, 5, 12, 29, 70, 169, 408, 985,
    In OEIS: - A000129 Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2). Also A069306 Number of 2 X n binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.
  • Sequence: 1, 3, 7, 17, 41, 99, 239, 577, 1393,
    In OEIS: - A001333 Numerators of continued fraction convergents to sqrt(2). Also A078057 Expansion of (1+x)/(1-2*x-x^2).
  • Sequence: 1, 4, 9, 22, 53, 128, 309, 746, 1801,
    In OEIS: - A048654 Generalized Pellian with second term equal to 4.
  • Sequence: 1, 5, 11, 27, 65, 157, 379, 915, 2209,
    In OEIS: - A048655 Generalized Pellian with second term equal to 5.
  • Sequence: 1, 6, 13, 32, 77, 186, 449, 1084, 2617,
    In OEIS: - A048693 Generalized Pellian with 2nd term equal to 6.
  • Sequence: 1, 7, 15, 37, 89, 215, 519, 1253, 3025,
    In OEIS: - A048694 Generalized Pellian with second term equal to 7.
  • Sequence: 1, 8, 17, 42, 101, 244, 589, 1422, 3433,
    In OEIS: - A048695 Generalized Pellian with second term equal to 8.
  • Sequence: 1, 9, 19, 47, 113, 273, 659, 1591, 3841,
    In OEIS: - A048696 Generalized Pellian with second term equal to 9.
  • Sequence: 1, 10, 21, 52, 125, 302, 729, 1760, 4249,
    In OEIS: - A048697 Generalized Pellian with second term equal to 10.
  • Sequence: 2, 2, 6, 14, 34, 82, 198, 478, 1154,
    In OEIS: - A002203 Companion Pell numbers: a(n) = 2a(n-1) + a(n-2).
  • Sequence: 2, 3, 8, 19, 46, 111, 268, 647, 1562,
    In OEIS: - A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).
  • Sequence: 5, 14, 33, 80, 193, 466, 1125, 2716,
    In OEIS: - A105082 G.f. (5+4x)/(1-2x-x^2).
• a(n) = 3a(n-1) + a(n-2).
  • Sequence: 1, 2, 7, 23, 76, 251, 829, 2738, 9043,
    In OEIS: - A052924 G.f.: (1-x)/(1-3*x-x^2).
  • Sequence: 1, 3, 10, 33, 109, 360, 1189, 3927,
    In OEIS: - A006190 a(n) = 3*a(n-1) + a(n-2). Also A020704 Pisot sequences E(3,10), P(3,10).
  • Sequence: 1, 4, 13, 43, 142, 469, 1549, 5116,
    In OEIS: - A003688 a(n) = 3*a(n-1) + a(n-2).
  • Sequence: 1, 5, 16, 53, 175, 578, 1909, 6305,
    In OEIS: - A006190 a(n) = A108300 a(n+2) = 3*a(n+1) + a(n), a(0) = 1, a(1) = 5.
  • Sequence: 2, 3, 11, 36, 119, 393, 1298, 4287,
    In OEIS: - A006497 a(n) = 3a(n-1) + a(n-2).
  • Sequence: 2, 7, 30, 127, 538, 2279, 9654, 40895,
    In OEIS: - A097924 Sequence relates numerators and denominators in the continued fraction convergents to sqrt(5).
• a(n) = 4a(n-1) + a(n-2).
  • Sequence: 1, 2, 9, 38, 161, 682, 2889,
    In OEIS: - A001077 Numerators of continued fraction convergents to sqrt(5).
  • Sequence: 1, 3, 13, 55, 233, 987, 4181,
    In OEIS: - A033887 Fibonacci(3n+1).
  • Sequence: 1, 4, 17, 72, 305, 1292, 5473,
    In OEIS: - A001076 Denominators of continued fraction convergents to sqrt(5).
  • Sequence: 1, 5, 21, 89, 377, 1597, 6765,
    In OEIS: - A015448 Generalized Fibonacci numbers: a(n) = 4*a(n-1) + a(n-2).
  • Sequence: 1, 6, 25, 106, 449, 1902, 8057,
    In OEIS: - A048875 Generalized Pellian with second term of 6.
  • Sequence: 1, 7, 29, 123, 521, 2207, 9349,
    In OEIS: - A048876 Generalized Pellian with second term of 7.
  • Sequence: 1, 8, 33, 140, 593, 2512, 10641,
    In OEIS: - A048877 Generalized Pellian with second term of 8.
  • Sequence: 1, 9, 37, 157, 665, 2817, 11933,
    In OEIS: - A048878 Generalized Pellian with second term of 9.
  • Sequence: 1, 10, 41, 174, 737, 3122, 13225,
    In OEIS: - A048879 Generalized Pellian with second term of 10.
  • Sequence: 2, 4, 18, 76, 322, 1364, 5778,
    In OEIS: - A014448 Even Lucas numbers: L(3n).
  • Sequence: 2, 8, 34, 144, 610, 2584, 10946,
    In OEIS: - A014445 Even Fibonacci numbers; or, Fibonacci_{3k}.
• a(n) = 5a(n-1) + a(n-2).
  • Sequence: 1, 4, 21, 109, 566, 2939, 15261, 79244,
    In OEIS: - A100237 Secondary diagonal of triangle A100235 divided by row number: a(n) = A100235(n+1,n)/(n+1) for n>=0.
  • Sequence: 1, 5, 26, 135, 701, 3640, 18901, 98145,
    In OEIS: - A052918 a(0)=1, a(1)=5, a(n+1) = 5*a(n) + a(n-1).
  • Sequence: 1, 6, 31, 161, 836, 4341, 22541,
    In OEIS: - A015449 Generalized Fibonacci numbers.
  • Sequence: 2, 5, 27, 140, 727, 3775, 19602, 101785,
    In OEIS: - A087130 a(n)=5*a(n-1)+a(n-2); a(0)=2, a(1)=5.
• a(n) = 6a(n-1) + a(n-2).
  • Sequence: 1, 1, 7, 43, 265, 1633, 10063,
    In OEIS: - A015451 a(n) = 6 a(n-1) + a(n-2).
  • Sequence: 1, 3, 19, 117, 721, 4443, 27379,
    In OEIS: - A005667 Numerators of continued fraction convergents to sqrt(10).
  • Sequence: 1, 6, 37, 228, 1405, 8658, 53353,
    In OEIS: - A005668 Denominators of continued fraction convergents to sqrt(10).
  • Sequence: 2, 6, 38, 234, 1442, 8886, 54758,
    In OEIS: - A085447 a(n) = 6*a(n-1) + a(n-2), starting with a(0)=2 and a(1)=6.
  • Sequence: 0, 2, 12, 74, 456, 2810, 17316,
    In OEIS: - A078469 Number of different compositions of the ladder graph L_n.
• a(n) = 7a(n-1) + a(n-2).
  • Sequence: 1, 1, 8, 57, 407, 2906, 20749, 148149,
    In OEIS: - A015453 Generalized Fibonacci numbers.
  • Sequence: 1, 7, 50, 357, 2549, 18200, 129949,
    In OEIS: - A054413 a(n)=7*a(n-1)+a(n-2).
  • Sequence: 2, 7, 51, 364, 2599, 18557, 132498,
    In OEIS: - A086902 a(n) = 7a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7, a(n) = [(7+sqrt(53))/2]^n + [(7-sqrt(53))/2]^n.
• a(n) = 8a(n-1) + a(n-2).
  • Sequence: 1, 1, 9, 73, 593, 4817, 39129, 317849,
    In OEIS: - A015454 Generalized Fibonacci numbers.
  • Sequence: 1, 4, 33, 268, 2177, 17684, 143649,
    In OEIS: - A041024 Numerators of continued fraction convergents to sqrt(17). Also A088317 a(n) = 8a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.
  • Sequence: 1, 8, 65, 528, 4289, 34840, 283009,
    In OEIS: - A041025 Denominators of continued fraction convergents to sqrt(17).
  • Sequence: 2, 8, 66, 536, 4354, 35368, 287298,
    In OEIS: - A086594 a(n)=8a(n-1)+a(n-2), starting with a(0)=2 and a(1)=8.
• a(n) = 9a(n-1) + a(n-2).
  • Sequence: 1, 1, 10, 91, 829, 7552, 68797, 626725,
    In OEIS: - A015455 Generalized Fibonacci numbers.
  • Sequence: 0, 1, 9, 82, 747, 6805, 61992, 564733,
    In OEIS: - A099371 Generalized Fibonacci sequence.
  • Sequence: 2, 9, 83, 756, 6887, 62739, 571538,
    In OEIS: - A087798 a(n) = 9a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 9, a(n) = [(9+sqrt(85))/2]^n + [(9-sqrt(85))/2]^n.
• a(n) = 10a(n-1) +