Generalizations of Fibonacci numbers

Using $F\_\{n-2\}\; =\; F\_n\; -\; F\_\{n-1\}$, one can extend the Fibonacci numbers to negative integers. So we get:

:... −8, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, 8, ...

and $F\_\{-n\}\; =\; (-1)^\{n\; +\; 1\}\; F\_n$.Triana, Juan. [http://www.viam.science.tsu.ge/others/ticmi/blt/vol23_1/3_triana.pdf Negafibonacci numbers via matrices. Bulletin of TICMI], 2019, pp. 19–24.

See also Negafibonacci coding.

There are a number of possible generalizations of the Fibonacci numbers which include the real numbers (and sometimes the complex numbers) in their domain. These each involve the golden ratio {{mvar|φ}}, and are based on Binet's formula

:$F\_n\; =\; \backslash frac\{\backslash varphi^n\; -\; (-\backslash varphi)^\{-n\}\}\{\backslash sqrt\{5\}\}.$

The analytic function

:$\backslash operatorname\{Fe\}(x)\; =\; \backslash frac\{\backslash varphi^x\; -\; \backslash varphi^\{-x\}\}\{\backslash sqrt\{5\}\}$

has the property that $\backslash operatorname\{Fe\}(n)\; =\; F\_n$ for even integers $n$.{{Cite web |date=2009-10-27 |title=What is a Fibonacci Number? -- from Harry J. Smith |url=http://geocities.com/hjsmithh/Fibonacc/FibWhat.html |access-date=2022-04-12 |website= |archive-url=https://web.archive.org/web/20091027103713/http://geocities.com/hjsmithh/Fibonacc/FibWhat.html |archive-date=27 October 2009 |url-status=dead}} Similarly, the analytic function:

:$\backslash operatorname\{Fo\}(x)\; =\; \backslash frac\{\backslash varphi^x\; +\; \backslash varphi^\{-x\}\}\{\backslash sqrt\{5\}\}$

satisfies $\backslash operatorname\{Fo\}(n)\; =\; F\_n$ for odd integers $n$.

Finally, putting these together, the analytic function

:$\backslash operatorname\{Fib\}(x)\; =\; \backslash frac\{\backslash varphi^x\; -\; \backslash cos(x\; \backslash pi)\backslash varphi^\{-x\}\}\{\backslash sqrt\{5\}\}$

satisfies $\backslash operatorname\{Fib\}(n)\; =\; F\_n$ for all integers $n$.{{MathWorld|title=Fibonacci Number|urlname=FibonacciNumber|author=Pravin Chandra and Eric W. Weisstein}}

Since $\backslash operatorname\{Fib\}(z\; +\; 2)\; =\; \backslash operatorname\{Fib\}(z\; +\; 1)\; +\; \backslash operatorname\{Fib\}(z)$ for all complex numbers $z$, this function also provides an extension of the Fibonacci sequence to the entire complex plane. Hence we can calculate the generalized Fibonacci function of a complex variable, for example,

:$\backslash operatorname\{Fib\}(3+4i)\; \backslash approx\; -5248.5\; -\; 14195.9\; i$

The term Fibonacci sequence is also applied more generally to any function $g$ from the integers to a field for which $g(n\; +\; 2)\; =\; g(n)\; +\; g(n\; +\; 1)$. These functions are precisely those of the form $g(n)\; =\; F(n)\; g(1)\; +\; F(n\; -\; 1)\; g(0)$, so the Fibonacci sequences form a vector space with the functions $F(n)$ and $F(n\; -\; 1)$ as a basis.

More generally, the range of $g$ may be taken to be any abelian group (regarded as a Z-module). Then the Fibonacci sequences form a 2-dimensional Z-module in the same way.

The 2-dimensional $\backslash mathbb\{Z\}$-module of Fibonacci integer sequences consists of all integer sequences satisfying $g(n\; +\; 2)\; =\; g(n)\; +\; g(n\; +\; 1)$. Expressed in terms of two initial values we have:

:$g(n)\; =\; F(n)g(1)\; +\; F(n-1)g(0)\; =\; g(1)\backslash frac\{\backslash varphi^n-(-\backslash varphi)^\{-n\}\}\{\backslash sqrt\; 5\}+g(0)\backslash frac\{\backslash varphi^\{n-1\}-(-\backslash varphi)^\{1-n\}\}\{\backslash sqrt\; 5\},$

where $\backslash varphi$ is the golden ratio.

The ratio between two consecutive elements converges to the golden ratio, except in the case of the sequence which is constantly zero and the sequences where the ratio of the two first terms is $(-\backslash varphi)^\{-1\}$.

The sequence can be written in the form

:$a\backslash varphi^n+b(-\backslash varphi)^\{-n\},$

in which $a\; =\; 0$ if and only if $b\; =\; 0$. In this form the simplest non-trivial example has $a\; =\; b\; =\; 1$, which is the sequence of Lucas numbers:

:$L\_n\; =\; \backslash varphi^n\; +\; (-\backslash varphi)^\{-\; n\}.$

We have $L\_1\; =\; 1$ and $L\_2\; =\; 3$. The properties include:

:$\backslash begin\{align\}$

\varphi^n &= \left(\frac{1+\sqrt{5}}{2}\right)^{\!n} = \frac{L(n)+F(n)\sqrt{5}}{2}, \\

L(n) &= F(n-1) + F(n+1).

\end{align}

Every nontrivial Fibonacci integer sequence appears (possibly after a shift by a finite number of positions) as one of the rows of the Wythoff array. The Fibonacci sequence itself is the first row, and a shift of the Lucas sequence is the second row.{{citation

| last = Morrison

| first = D. R.

| contribution = A Stolarsky array of Wythoff pairs

| location = Santa Clara, CA

| pages = 134–136

| publisher = The Fibonacci Association

| title = A Collection of Manuscripts Related to the Fibonacci Sequence

| url = http://www.math.ucsb.edu/~drm/papers/stolarsky.pdf

| year = 1980

| access-date = 2012-07-15

| archive-url = https://web.archive.org/web/20160304024024/http://www.math.ucsb.edu/~drm/papers/stolarsky.pdf

| archive-date = 2016-03-04

| url-status = dead

}}.

See also Fibonacci integer sequences modulo n.

A different generalization of the Fibonacci sequence is the Lucas sequences of the kind defined as follows:

:$\backslash begin\{align\}$

U(0) &= 0 \\

U(1) &= 1 \\

U(n + 2) &= P U(n + 1) - Q U(n),

\end{align}

where the normal Fibonacci sequence is the special case of $P\; =\; 1$ and $Q\; =\; -1$. Another kind of Lucas sequence begins with $V(0)\; =\; 2$, $V(1)\; =\; P$. Such sequences have applications in number theory and primality proving.

When $Q\; =\; -1$, this sequence is called {{mvar|P}}-Fibonacci sequence, for example, Pell sequence is also called 2-Fibonacci sequence.

The 3-Fibonacci sequence is

:0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, 467280, 1543321, 5097243, 16835050, 55602393, 183642229, 606529080, ... {{OEIS|id=A006190}}

The 4-Fibonacci sequence is

:0, 1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020, 1762289, 7465176, 31622993, 133957148, 567451585, 2403763488, ... {{OEIS|id=A001076}}

The 5-Fibonacci sequence is

:0, 1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275, 13741001, 71351280, 370497401, 1923838285, 9989688826, ... {{OEIS|id=A052918}}

The 6-Fibonacci sequence is

:0, 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, 12484830, 76934989, 474094764, 2921503573, 18003116202, ... {{OEIS|id=A005668}}

The {{mvar|n}}-Fibonacci constant is the ratio toward which adjacent $n$-Fibonacci numbers tend; it is also called the {{mvar|n}}th metallic mean, and it is the only positive root of $x^2\; -\; nx\; -\; 1\; =\; 0$. For example, the case of $n\; =\; 1$ is $\backslash frac\{1\; +\; \backslash sqrt\{5\}\}\{2\}$, or the golden ratio, and the case of $n\; =\; 2$ is $1\; +\; \backslash sqrt\{2\}$, or the silver ratio. Generally, the case of $n$ is $\backslash frac\{n\; +\; \backslash sqrt\{n^2\; +\; 4\}\}\{2\}$.{{citation needed|reason=as the IP got it wrong, it needs a reference. The quadratic formula does not fall under WP:CALC.|date=September 2014}}

Generally, $U(n)$ can be called {{math|(P,−Q)}}-Fibonacci sequence, and {{math|V(n)}} can be called {{math|(P,−Q)}}-Lucas sequence.

The (1,2)-Fibonacci sequence is

:0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, ... {{OEIS|id=A001045}}

The (1,3)-Fibonacci sequence is

:1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, 14209, 32689, 75316, 173383, 399331, 919480, 2117473, 4875913, 11228332, 25856071, 59541067, ... {{OEIS|id=A006130}}

The (2,2)-Fibonacci sequence is

:0, 1, 2, 6, 16, 44, 120, 328, 896, 2448, 6688, 18272, 49920, 136384, 372608, 1017984, 2781184, 7598336, 20759040, 56714752, ... {{OEIS|id=A002605}}

The (3,3)-Fibonacci sequence is

:0, 1, 3, 12, 45, 171, 648, 2457, 9315, 35316, 133893, 507627, 1924560, 7296561, 27663363, 104879772, 397629405, 1507527531, 5715470808, ... {{OEIS|id=A030195}}

A Fibonacci sequence of order {{mvar|n}} is an integer sequence in which each sequence element is the sum of the previous $n$ elements (with the exception of the first $n$ elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2. The cases $n\; =\; 3$ and $n\; =\; 4$ have been thoroughly investigated. The number of compositions of nonnegative integers into parts that are at most $n$ is a Fibonacci sequence of order $n$. The sequence of the number of strings of 0s and 1s of length $m$ that contain at most $n$ consecutive 0s is also a Fibonacci sequence of order $n$.

These sequences, their limiting ratios, and the limit of these limiting ratios, were investigated by Mark Barr in 1913.{{cite book

|last=Gardner|first=Martin|author-link=Martin Gardner

|page=101|title=The Scientific American Book of Mathematical Puzzles and Diversions, Vol. II

|publisher=Simon and Schuster|year=1961}}

The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are:

:0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, … {{OEIS|A000073}}

The series was first described formally by Agronomof{{cite journal | last1 = Tuenter | first1 = Hans J. H. | doi = 10.1080/00029890.2023.2231796 | title = In Search of Comrade Agronomof: Some Tribonacci History | journal = The American Mathematical Monthly | volume = 130 | issue = 8 | pages = 708–719 |date=October 2023| mr=4645497 }} in 1914,{{Cite journal|last=Agronomof|first=M.|year=1914|title=Sur une suite récurrente.|journal=Mathesis|volume=4|pages=125–126}} but its first unintentional use is in the Origin of Species by Charles R. Darwin. In the example of illustrating the growth of elephant population, he relied on the calculations made by his son, George H. Darwin.{{Cite journal | last1 = Podani | first1 = János | last2 = Kun | first2 = Ádám | last3 = Szilágyi | first3 = András |year = 2018 | title = How Fast Does Darwin's Elephant Population Grow? | journal = Journal of the History of Biology | volume = 51 | issue = 2 | pages = 259–281 | doi = 10.1007/s10739-017-9488-5| pmid = 28726021 | s2cid = 3988121 | url = http://real.mtak.hu/72652/1/Podani_etal_DarwinElephants.pdf }} The term tribonacci was suggested by Feinberg in 1963.{{Cite journal | last = Feinberg | first = M. | year = 1963 | title = Fibonacci-Tribonacci. | journal = Fibonacci Quarterly | volume = 1 | pages = 71–74 }}

The tribonacci constant

:$\backslash frac\{1+\backslash sqrt[3]\{19+3\backslash sqrt\{33\}\}+\backslash sqrt[3]\{19-3\backslash sqrt\{33\}\}\}\{3\}\; =\; \backslash frac\{1+4\backslash cosh\backslash left(\backslash frac\{1\}\{3\}\backslash cosh^\{-1\}\backslash left(2+\backslash frac\{3\}\{8\}\backslash right)\backslash right)\}\{3\}\; \backslash approx\; 1.839286755214161,$ {{OEIS|A058265}}

is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial $x^3\; -\; x^2\; -\; x\; -\; 1\; =\; 0$, and also satisfies the equation $x\; +\; x^\{-3\}\; =\; 2$. It is important in the study of the snub cube.

File:TRIBONACCI.jpg

The reciprocal of the tribonacci constant, expressed by the relation $\backslash xi^3\; +\; \backslash xi^2\; +\; \backslash xi\; =\; 1$, can be written as:

:$\backslash xi\; =\; \backslash frac\{\backslash sqrt[3]\{17+3\backslash sqrt\{33\}\}\; -\; \backslash sqrt[3]\{-17+3\backslash sqrt\{33\}\}\; -\; 1\}\{3\}\; =\; \backslash frac\{3\}\{1\; +\; \backslash sqrt[3]\{19\; +\; 3\backslash sqrt\{33\}\}\; +\; \backslash sqrt[3]\{19-3\backslash sqrt\{33\}\}\}\; \backslash approx\; 0.543689012.$ {{OEIS|A192918}}

The tribonacci numbers are also given by[http://www.plouffe.fr Simon Plouffe, 1993]

:$T(n)\; =\; \backslash left\backslash lfloor\; 3b\backslash ,\; \backslash frac\{\backslash left(\backslash frac\{1\}\{3\}\; \backslash left(\; a\_\{+\}\; +\; a\_\{-\}\; +\; 1\backslash right)\backslash right)^n\}\{b^2-2b+4\}\; \backslash right\backslash rceil$

where $\backslash lfloor\; \backslash cdot\; \backslash rceil$ denotes the nearest integer function and

:$\backslash begin\{align\}$

a_{\pm} &= \sqrt[3]{19 \pm 3 \sqrt{33}}\,, \\

b &= \sqrt[3]{586 + 102 \sqrt{33}}\,.

\end{align}

The tetranacci numbers start with four predetermined terms, each term afterwards being the sum of the preceding four terms. The first few tetranacci numbers are:

:0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, … {{OEIS|A000078}}

The tetranacci constant is the ratio toward which adjacent tetranacci numbers tend. It is a root of the polynomial $x^4\; -\; x^3\; -\; x^2\; -\; x\; -\; 1\; =\; 0$, approximately 1.927561975482925 {{OEIS|A086088}}, and also satisfies the equation $x\; +\; x^\{-4\}\; =\; 2$.

The tetranacci constant can be expressed in terms of radicals by the following expression:{{Cite journal|last=Wolfram|first=D.A.|date=1998|title=Solving Generalized Fibonacci Recurrences|url=http://www.fq.math.ca/Scanned/36-2/wolfram.pdf|journal=Fib. Quart.}}

:$x\; =\; \backslash frac\{1\}\{4\}\backslash !\backslash left(1+\backslash sqrt\{u\}+\backslash sqrt\{11-u+\backslash frac\{26\}\{\backslash sqrt\{u\}\}\}\backslash ,\backslash right)$

where,

:$u\; =\; \backslash frac\{1\}\{3\}\backslash left(11-56\backslash sqrt[3]\{\backslash frac\{2\}\{-65+3\backslash sqrt\{1689\}\}\}+2\backslash cdot2^\{\backslash frac\{2\}\{3\}\}\backslash sqrt[3]\{-65+3\backslash sqrt\{1689\}\}\backslash right)$

and $u$ is the real root of the cubic equation $u^3-11u^2+115u-169.$

Pentanacci, hexanacci, heptanacci, octanacci and enneanacci numbers have been computed.

Pentanacci numbers:

:0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, … {{OEIS|A001591}}

The pentanacci constant is the ratio toward which adjacent pentanacci numbers tend.

It is a root of the polynomial $x^5\; -\; x^4\; -\; x^3\; -\; x^2\; -\; x\; -\; 1\; =\; 0$, approximately 1.965948236645485 {{OEIS|A103814}}, and also satisfies the equation $x\; +\; x^\{-5\}\; =\; 2$.

Hexanacci numbers:

:0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, … {{OEIS|A001592}}

The hexanacci constant is the ratio toward which adjacent hexanacci numbers tend.

It is a root of the polynomial $x^6\; -\; x^5\; -\; x^4\; -\; x^3\; -\; x^2\; -\; x\; -\; 1\; =\; 0$, approximately 1.98358284342 {{OEIS| A118427}}, and also satisfies the equation $x\; +\; x^\{-6\}\; =\; 2$.

Heptanacci numbers:

:0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, … {{OEIS|A122189}}

The heptanacci constant is the ratio toward which adjacent heptanacci numbers tend.

It is a root of the polynomial $x^7\; -\; x^6\; -\; x^5\; -\; x^4\; -\; x^3\; -\; x^2\; -\; x\; -\; 1\; =\; 0$, approximately 1.99196419660 {{OEIS|A118428}}, and also satisfies the equation $x\; +\; x^\{-7\}\; =\; 2$.

Octanacci numbers:

:0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, ... {{OEIS|A079262}}

Enneanacci numbers:

:0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, ... {{OEIS|A104144}}

The limit of the ratio of successive terms of an $n$-nacci series tends to a root of the equation $x\; +\; x^\{-n\}\; =\; 2$ ({{OEIS2C|A103814}}, {{OEIS2C|A118427}}, {{OEIS2C|A118428}}).

An alternate recursive formula for the limit of ratio $r$ of two consecutive $n$-nacci numbers can be expressed as

:$r=\backslash sum\_\{k=0\}^\{n-1\}r^\{-k\}$.

The special case $n\; =\; 2$ is the traditional Fibonacci series yielding the golden section $\backslash varphi\; =\; 1\; +\; \backslash frac\{1\}\{\backslash varphi\}$.

The above formulas for the ratio hold even for $n$-nacci series generated from arbitrary numbers. The limit of this ratio is 2 as $n$ increases. An "infinacci" sequence, if one could be described, would after an infinite number of zeroes yield the sequence

:[..., 0, 0, 1,] 1, 2, 4, 8, 16, 32, …

which are simply the powers of two.

The limit of the ratio for any $n\; >\; 0$ is the positive root $r$ of the characteristic equation

:$x^n\; -\; \backslash sum\_\{i\; =\; 0\}^\{n-1\}\; x^i\; =\; 0.$

The root $r$ is in the interval . The negative root of the characteristic equation is in the interval (−1, 0) when $n$ is even. This root and each complex root of the characteristic equation has modulus .

A series for the positive root $r$ for any $n\; >\; 0$ is

:$2\; -\; 2\backslash sum\_\{i\; >\; 0\}\; \backslash frac\{1\}\{i\}\backslash binom\{(n+1)i\; -2\}\{i-1\}\backslash frac\{1\}\{2^\{(n+1)i\}\}.$

There is no solution of the characteristic equation in terms of radicals when {{math|5 ≤ n ≤ 11}}.

The {{mvar|k}}th element of the {{mvar|n}}-nacci sequence is given by

:$F\_k^\{(n)\}\; =\; \backslash left\backslash lfloor\; \backslash frac\{r^\{k-1\}\; (r-1)\}\{(n+1)r-2n\}\backslash right\backslash rceil\backslash !,$

where $\backslash lfloor\; \backslash cdot\; \backslash rceil$ denotes the nearest integer function and $r$ is the $n$-nacci constant, which is the root of $x\; +\; x^\{-n\}\; =\; 2$ nearest to 2.

A coin-tossing problem is related to the $n$-nacci sequence. The probability that no $n$ consecutive tails will occur in $m$ tosses of an idealized coin is $\backslash frac\{1\}\{2^m\}F^\{(n)\}\_\{m\; +\; 2\}$.{{MathWorld|title=Coin Tossing|urlname=CoinTossing|author=Eric W. Weisstein}}

{{main|Fibonacci word}}

In analogy to its numerical counterpart, the Fibonacci word is defined by:

:$F\_n\; :=\; F(n):=\; \backslash begin\{cases\}$

\text{b} & n = 0; \\

\text{a} & n = 1; \\

F(n-1)+F(n-2) & n > 1. \\

\end{cases}

where $+$ denotes the concatenation of two strings. The sequence of Fibonacci strings starts:

:{{not a typo|b, a, ab, aba, abaab, abaababa, abaababaabaab,}} … {{OEIS|id=A106750}}

The length of each Fibonacci string is a Fibonacci number, and similarly there exists a corresponding Fibonacci string for each Fibonacci number.

Fibonacci strings appear as inputs for the worst case in some computer algorithms.

If "a" and "b" represent two different materials or atomic bond lengths, the structure corresponding to a Fibonacci string is a Fibonacci quasicrystal, an aperiodic quasicrystal structure with unusual spectral properties.

A convolved Fibonacci sequence is obtained applying a convolution operation to the Fibonacci sequence one or more times. Specifically, defineV. E. Hoggatt, Jr. and M. Bicknell-Johnson, [http://www.fq.math.ca/Scanned/15-2/hoggatt1.pdf "Fibonacci Convolution Sequences"], Fib. Quart., 15 (1977), pp. 117-122.

:$F\_n^\{(0)\}=F\_n$

and

:$F\_n^\{(r+1)\}=\backslash sum\_\{i=0\}^n\; F\_i\; F\_\{n-i\}^\{(r)\}$

The first few sequences are

:$r\; =\; 1$: 0, 0, 1, 2, 5, 10, 20, 38, 71, … {{OEIS|A001629}}.

:$r\; =\; 2$: 0, 0, 0, 1, 3, 9, 22, 51, 111, … {{OEIS|A001628}}.

:$r\; =\; 3$: 0, 0, 0, 0, 1, 4, 14, 40, 105, … {{OEIS|A001872}}.

The sequences can be calculated using the recurrence

:$F\_\{n+1\}^\{(r+1)\}=F\_n^\{(r+1)\}+F\_\{n-1\}^\{(r+1)\}+F\_n^\{(r)\}$

The generating function of the $r$th convolution is

:$s^\{(r)\}(x)=\backslash sum\_\{k=0\}^\{\backslash infty\}\; F^\{(r)\}\_n\; x^n=\backslash left(\backslash frac\{x\}\{1-x-x^2\}\backslash right)^r.$

The sequences are related to the sequence of Fibonacci polynomials by the relation

:$F\_n^\{(r)\}=r!\; F\_n^\{(r)\}(1)$

where $F^\{(r)\}\_n(x)$ is the $r$th derivative of $F\_n(x)$. Equivalently, $F^\{(r)\}\_n$ is the coefficient of $(x\; -\; 1)^r$ when $F\_x(x)$ is expanded in powers of $(x\; -\; 1)$.

The first convolution, $F^\{(1)\}\_n$ can be written in terms of the Fibonacci and Lucas numbers as

:$F\_n^\{(1)\}=\backslash frac\{nL\_n-F\_n\}\{5\}$

and follows the recurrence

:$F\_\{n+1\}^\{(1)\}=2F\_n^\{(1)\}+F\_\{n-1\}^\{(1)\}-2F\_\{n-2\}^\{(1)\}-F\_\{n-3\}^\{(1)\}.$

Similar expressions can be found for $r\; >\; 1$ with increasing complexity as $r$ increases. The numbers $F^\{(1)\}\_n$ are the row sums of Hosoya's triangle.

As with Fibonacci numbers, there are several combinatorial interpretations of these sequences. For example $F^\{(1)\}\_n$ is the number of ways $n\; -\; 2$ can be written as an ordered sum involving only 0, 1, and 2 with 0 used exactly once. In particular $F^\{(1)\}\_4\; =\; 5$ and 2 can be written {{nowrap|0 + 1 + 1}}, {{nowrap|0 + 2}}, {{nowrap|1 + 0 + 1}}, {{nowrap|1 + 1 + 0}}, {{nowrap|2 + 0}}.{{Cite OEIS|sequencenumber=A001629}}

The Fibonacci polynomials are another generalization of Fibonacci numbers.

The Padovan sequence is generated by the recurrence $P(n)\; =\; P(n\; -\; 2)\; +\; P(n\; -\; 3)$.

The Narayana's cows sequence is generated by the recurrence $N(k)\; =\; N(k\; -\; 1)\; +\; N(k\; -\; 3)$.

A random Fibonacci sequence can be defined by tossing a coin for each position $n$ of the sequence and taking $F(n)\; =\; F(n\; -\; 1)\; +\; F(n\; -\; 2)$ if it lands heads and $F(n)\; =\; F(n\; -\; 1)\; -\; F(n\; -\; 2)$ if it lands tails. Work by Furstenberg and Kesten guarantees that this sequence almost surely grows exponentially at a constant rate: the constant is independent of the coin tosses and was computed in 1999 by Divakar Viswanath. It is now known as Viswanath's constant.

A repfigit, or Keith number, is an integer such that, when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 {{nowrap|(4, 7, 11, 18, 29, 47)}} reaches 47. A repfigit can be a tribonacci sequence if there are 3 digits in the number, a tetranacci number if the number has four digits, etc. The first few repfigits are:

:14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, … {{OEIS|A007629}}

Since the set of sequences satisfying the relation $S(n)\; =\; S(n\; -\; 1)\; +\; S(n\; -\; 2)$ is closed under termwise addition and under termwise multiplication by a constant, it can be viewed as a vector space. Any such sequence is uniquely determined by a choice of two elements, so the vector space is two-dimensional. If we abbreviate such a sequence as $(S(0),\; S(1))$, the Fibonacci sequence $F(n)\; =\; (0,\; 1)$ and the shifted Fibonacci sequence $F(n\; -\; 1)\; =\; (1,\; 0)$ are seen to form a canonical basis for this space, yielding the identity:

:$S(n)\; =\; S(0)\; F(n-1)\; +\; S(1)\; F(n)$

for all such sequences {{mvar|S}}. For example, if {{mvar|S}} is the Lucas sequence {{nowrap|2, 1, 3, 4, 7, 11, ...}}, then we obtain

:$L(n)\; =\; 2F(n-1)\; +\; F(n)$.

We can define the {{mvar|N}}-generated Fibonacci sequence (where {{mvar|N}} is a positive rational number): if

:$N\; =\; 2^\{a\_1\}\backslash cdot\; 3^\{a\_2\}\backslash cdot\; 5^\{a\_3\}\backslash cdot\; 7^\{a\_4\}\backslash cdot\; 11^\{a\_5\}\backslash cdot\; 13^\{a\_6\}\backslash cdot\; \backslash ldots\; \backslash cdot\; p\_r^\{a\_r\},$

where {{mvar|p_r}} is the {{mvar|r}}th prime, then we define

:$F\_N(n)\; =\; a\_1F\_N(n-1)\; +\; a\_2F\_N(n-2)\; +\; a\_3F\_N(n-3)\; +\; a\_4F\_N(n-4)\; +\; a\_5F\_N(n-5)\; +\; ...$

If $n\; =\; r\; -\; 1$, then $F\_N(n)\; =\; 1$, and if , then $F\_N(n)\; =\; 0$.{{citation needed|reason=has this been defined outside of this article?|date=June 2019}}

:

The semi-Fibonacci sequence {{OEIS|A030067}} is defined via the same recursion for odd-indexed terms $a(2n+1)\; =\; a(2n)\; +\; a(2n-1)$ and $a(1)\; =\; 1$, but for even indices $a(2n)\; =\; a(n)$, $n\; \backslash ge\; 1$. The bisection {{OEIS link|id=A030068}} of odd-indexed terms $s(n)\; =\; a(2n-1)$ therefore verifies $s(n+1)\; =\; s(n)\; +\; a(n)$ and is strictly increasing. It yields the set of the semi-Fibonacci numbers

: 1, 2, 3, 5, 6, 9, 11, 16, 17, 23, 26, 35, 37, 48, 53, 69, 70, 87, 93, 116, 119, 145, 154, ... {{OEIS|A030068}}

which occur as $s(n)\; =\; a(2^k(2n-1)),\; k=0,1,...\backslash ,\; .$

{{reflist|2}}

• {{springer|title=Tribonacci number|id=p/t130190}}

{{Fibonacci}}

Category:Fibonacci numbers