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# Recurrences with nonconstant 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:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=n\,a_{n-1},\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=2n^{2}\,a_{n-1},\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,a_{n-1},\quad n\geq 1.}$
##### Homogeneous linear recurrences (of order 2) with nonconstant coefficients

Examples:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,a_{n-1}+n\,a_{n-2},\quad n\geq 1.}$

#### 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:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=n\,a_{n-1}+1,\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=2n^{2}\,a_{n-1}+n,\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,a_{n-1}+2^{n},\quad n\geq 1.}$
##### Non-homogenuous linear recurrences (of order 2) with nonconstant coefficients

Examples:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,a_{n-1}+2^{n}\,a_{n-2}+n,\quad n\geq 1.}$

### 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

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#### 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:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=n\,{a_{n-1}}^{2},\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=2n^{2}\,{a_{n-1}}^{2},\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,{a_{n-1}}^{2},\quad n\geq 1.}$
##### Homogenuous quadratic recurrences (of order 2) with nonconstant coefficients

Examples:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,{a_{n-1}}^{2}+2^{n}\,a_{n-2},\quad n\geq 1.}$

#### 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:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=n\,{a_{n-1}}^{2}+1,\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=2n^{2}\,{a_{n-1}}^{2}+n,\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,{a_{n-1}}^{2}+2^{n},\quad n\geq 1.}$
##### Non-homogenuous quadratic recurrences (of order 2) with nonconstant coefficients

Examples:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,{a_{n-1}}^{2}+2^{n}\,a_{n-2}+n,\quad n\geq 1.}$

### 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:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=n\,{a_{n-1}}^{3},\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=2n^{2}\,{a_{n-1}}^{3},\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,{a_{n-1}}^{3},\quad n\geq 1.}$
##### Homogenuous cubic recurrences (of order 2) with nonconstant coefficients

Examples:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,{a_{n-1}}^{3}+2^{n}\,{a_{n-2}}^{2},\quad n\geq 1.}$

#### 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:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=n\,{a_{n-1}}^{3}+1,\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=2n^{2}\,{a_{n-1}}^{3}+n,\quad n\geq 1.}$

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,{a_{n-1}}^{3}+2^{n},\quad n\geq 1.}$
##### Non-homogenuous cubic recurrences (of order 2) with nonconstant coefficients

Examples:

${\displaystyle a_{0}:=1;}$
${\displaystyle a_{n}:=3^{n}\,{a_{n-1}}^{3}+2^{n}\,{a_{n-2}}^{2}+n,\quad n\geq 1.}$