Editing Fibonacci sequence (section)

== Generalizations ==
{{Main|Generalizations of Fibonacci numbers}}

The Fibonacci sequence is one of the simplest and earliest known sequences defined by a [[recurrence relation]], and specifically by a linear [[difference equation]]. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a [[linear recurrence with constant coefficients|homogeneous linear difference equation with constant coefficients]].

Some specific examples that are close, in some sense, to the Fibonacci sequence include:
* Generalizing the index to negative integers to produce the [[negafibonacci]] numbers.
* Generalizing the index to [[real number]]s using a modification of Binet's formula.<ref name="MathWorld" />
* Starting with other integers. [[Lucas number]]s have {{math|1=''L''<sub>1</sub> = 1}}, {{math|1=''L''<sub>2</sub> = 3}}, and {{math|1=''L<sub>n</sub>'' = ''L''<sub>''n''−1</sub> + ''L''<sub>''n''−2</sub>}}. [[Primefree sequence]]s use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.
* Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The [[Pell number]]s have {{math|1=''P<sub>n</sub>'' = 2''P''<sub>''n''−1</sub> + ''P''<sub>''n''−2</sub>}}.  If the coefficient of the preceding value is assigned a variable value {{mvar|x}}, the result is the sequence of [[Fibonacci polynomials]].
* Not adding the immediately preceding numbers. The [[Padovan sequence]] and [[Perrin number]]s have {{math|1=''P''(''n'') = ''P''(''n'' − 2) + ''P''(''n'' − 3)}}.
* Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as ''n-Step Fibonacci numbers''.<ref>{{citation
 | last1 = Lü | first1 = Kebo
 | last2 = Wang | first2 = Jun
 | journal = Utilitas Mathematica
 | mr = 2278830
 | pages = 169–177
 | title = {{mvar|k}}-step Fibonacci sequence modulo {{mvar|m}}
 | url = https://utilitasmathematica.com/index.php/Index/article/view/410
 | volume = 71
 | year = 2006}}</ref>