Recurrences with nonconstant coefficients

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This article is concerned with recurrences with nonconstant coefficients, as opposed to recurrences with constant coefficients.

Recurrences with nonconstant coefficients

Main article page: Recurrence relations with nonconstant coefficients

Linear recurrences with nonconstant coefficients

Main article page: Linear recurrence relations with nonconstant coefficients

Homogeneous linear recurrences with nonconstant coefficients

Main article page: Homogeneous linear recurrence relations with nonconstant coefficients

Homogeneous linear recurrences (of order 1) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=n\,a_{n-1},\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=2n^{2}\,a_{n-1},\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=3^{n}\,a_{n-1},\quad n\geq 1.$

Homogeneous linear recurrences (of order 2) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=3^{n}\,a_{n-1}+n\,a_{n-2},\quad n\geq 1.$

Non-homogeneous linear recurrences with nonconstant coefficients

Main article page: Non-homogeneous linear recurrence relations with nonconstant coefficients

Non-homogenuous linear recurrences (of order 1) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=n\,a_{n-1}+1,\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=2n^{2}\,a_{n-1}+n,\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=3^{n}\,a_{n-1}+2^{n},\quad n\geq 1.$

Non-homogenuous linear recurrences (of order 2) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=3^{n}\,a_{n-1}+2^{n}\,a_{n-2}+n,\quad n\geq 1.$

Quadratic recurrences with nonconstant coefficients

Main article page: Quadratic recurrence relations with nonconstant coefficients

Bilinear recurrence relations with nonconstant coefficients

Main article page: Bilinear recurrence relations with nonconstant coefficients

Homogeneous bilinear recurrence relations with nonconstant coefficients

(...)

Non-homogeneous bilinear recurrence relations with nonconstant coefficients

(...)

Homogeneous quadratic recurrences with nonconstant coefficients

Main article page: Homogeneous quadratic recurrence relations with nonconstant coefficients

Homogenuous quadratic recurrences (of order 1) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=n\,{a_{n-1}}^{2},\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=2n^{2}\,{a_{n-1}}^{2},\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=3^{n}\,{a_{n-1}}^{2},\quad n\geq 1.$

Homogenuous quadratic recurrences (of order 2) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=3^{n}\,{a_{n-1}}^{2}+2^{n}\,a_{n-2},\quad n\geq 1.$

Non-homogeneous quadratic recurrences with nonconstant coefficients

Main article page: Non-homogeneous quadratic recurrence relations with nonconstant coefficients

Non-homogenuous quadratic recurrences (of order 1) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=n\,{a_{n-1}}^{2}+1,\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=2n^{2}\,{a_{n-1}}^{2}+n,\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=3^{n}\,{a_{n-1}}^{2}+2^{n},\quad n\geq 1.$

Non-homogenuous quadratic recurrences (of order 2) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=3^{n}\,{a_{n-1}}^{2}+2^{n}\,a_{n-2}+n,\quad n\geq 1.$

Cubic recurrences with nonconstant coefficients

Main article page: Cubic recurrence relations with nonconstant coefficients

Homogeneous cubic recurrences with nonconstant coefficients

Main article page: Homogeneous cubic recurrence relations with nonconstant coefficients

Homogenuous cubic recurrences (of order 1) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=n\,{a_{n-1}}^{3},\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=2n^{2}\,{a_{n-1}}^{3},\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=3^{n}\,{a_{n-1}}^{3},\quad n\geq 1.$

Homogenuous cubic recurrences (of order 2) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=3^{n}\,{a_{n-1}}^{3}+2^{n}\,{a_{n-2}}^{2},\quad n\geq 1.$

Non-homogeneous cubic recurrences with nonconstant coefficients

Main article page: Non-homogeneous cubic recurrence relations with nonconstant coefficients

Non-homogenuous cubic recurrences (of order 1) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=n\,{a_{n-1}}^{3}+1,\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=2n^{2}\,{a_{n-1}}^{3}+n,\quad n\geq 1.$

$a_{0}:=1;$

$a_{n}:=3^{n}\,{a_{n-1}}^{3}+2^{n},\quad n\geq 1.$

Non-homogenuous cubic recurrences (of order 2) with nonconstant coefficients

Examples:

$a_{0}:=1;$

$a_{n}:=3^{n}\,{a_{n-1}}^{3}+2^{n}\,{a_{n-2}}^{2}+n,\quad n\geq 1.$

See also

• Homogeneous recurrence
• Non-homogeneous recurrence

• N th-order recurrence