Recurrence relations with constant coefficients

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

$(...)\,$

Linear recurrence relations with constant coefficients

Cf. Index entries for sequences related to linear recurrences with constant coefficients.

Homogeneous linear recurrence relations with constant coefficients

An order $\scriptstyle k\,$ 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 $\scriptstyle k\,$ coefficients $\scriptstyle c_{i}\,(\forall i)\,$ are constants.

Nonhomogeneous linear recurrence relations with constant coefficients

An order $\scriptstyle k\,$ 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 $\scriptstyle f(n)\,\neq \,0\,$ and the $\scriptstyle k\,$ coefficients $\scriptstyle c_{i}\,(\forall i)\,$ are constants.

Quadratic recurrence relations with constant coefficients

Homogeneous quadratic recurrence relations with constant coefficients

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Nonhomogeneous quadratic recurrence relations with constant coefficients

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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 $\scriptstyle 1+{\lfloor k/2\rfloor }\,$ coefficients $\scriptstyle d_{i}\,(\forall i)\,$ are constants. (Note that there are no squared $\scriptstyle a(j)\,(\forall j)\,$.)

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