Recursive Sequences


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:


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

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

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

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

Sequences:


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 :

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

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

Sequences:


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

Common properties for these sequences:

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

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

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

Sequences:


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

Common properties for sequences starting 1, d+1:

Common properties for sequences starting 2, d:

Common properties for sequences with positive d = 2k.

Sequences:


a(n) = d * a(n-1). Geometric progressions.

Common properties:

Sequences:


a(n) = a(n-1) + d. Arithmetic progressions.

Sequences:


a(n) = a(n-1). Constants.

Sequences:


Last revised March 2007