This site is supported by donations to The OEIS Foundation.

# Recurrence relations with constant coefficients

(Redirected from Homogeneous linear recurrence relations with constant coefficients)
Jump to: navigation, search

This article page is a stub, please help by expanding it.

A recurrence relation with constant coefficients of degree $d\,$ is an equation of the form

$(...)\,$ ## Linear recurrence relations with constant coefficients

### Homogeneous linear recurrence relations with constant coefficients

An order $k\,$ homogeneous linear recurrence relation with constant coefficients is an equation of the form

$\sum _{i=0}^{k}c_{i}\,a(n-i)=c_{0}\,a(n)+c_{1}\,a(n-1)+c_{2}\,a(n-2)+\cdots +c_{k}\,a(n-k)=0,\,$ where the $k\,$ coefficients $c_{i}\,(\forall i)\,$ are constants.

### Nonhomogeneous linear recurrence relations with constant coefficients

An order $k\,$ nonhomogeneous linear recurrence relation with constant coefficients is an equation of the form

$\left(\sum _{i=0}^{k}c_{i}\,a(n-i)\right)+f(n)={\Bigg (}c_{0}\,a(n)+c_{1}\,a(n-1)+c_{2}\,a(n-2)+\cdots +c_{k}\,a(n-k){\Bigg )}+f(n)=0,\,$ where $f(n)\,\neq \,0\,$ and the $k\,$ coefficients $c_{i}\,(\forall i)\,$ are constants.

## Quadratic recurrence relations with constant coefficients

(...)

(...)

### Bilinear recurrence relations with constant coefficients

#### Homogeneous bilinear recurrence relations with constant coefficients

A homogeneous bilinear recurrence relation with constant coefficients is an equation of the form

$\sum _{i=0}^{\lfloor k/2\rfloor }d_{i}\,a(n-i)\,a(n-k+i)=0,\,$ where the $1+{\lfloor k/2\rfloor }\,$ coefficients $d_{i}\,(\forall i)\,$ are constants. (Note that there are no squared $a(j)\,(\forall j)\,$ .)

#### Nonhomogeneous bilinear recurrence relations with constant coefficients

A nonhomogeneous bilinear recurrence relation with constant coefficients is an equation of the form

$\sum _{i=0}^{\lfloor k/2\rfloor }d_{i}\,a(n-i)\,a(n-k+i)+\left(\sum _{i=0}^{k}c_{i}\,a(n-i)\right)+f(n)=0,\,$ where either $\sum _{i=0}^{k}|c_{i}|\,\neq \,0\,$ or $f(n)\,\neq \,0\,$ and the $1+{\lfloor k/2\rfloor }\,$ coefficients $d_{i}\,(\forall i)\,$ and $c_{i}\,(\forall i)\,$ and are constants. (Note that there are no squared $a(j)\,(\forall j)\,$ .)