Generalizations of Fibonacci numbers

In mathematics, the Fibonacci numbers form a sequence defined recursively by:

$F_{n}={\begin{cases}0&n=0\\1&n=1\\F_{n-1}+F_{n-2}&n>1\end{cases}}$

That is, after two starting values, each number is the sum of the two preceding numbers.

The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers.

Extension to negative integers

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 $F_{-n}=(-1)^{n+1}F_{n}$.

See also Negafibonacci coding.

Extension to all real or complex numbers

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 φ, and are based on Binet's formula

$F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.$

The analytic function

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

has the property that $\operatorname {Fe} (n)=F_{n}$ for even integers $n$. Similarly, the analytic function:

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

satisfies $\operatorname {Fo} (n)=F_{n}$ for odd integers $n$.

Finally, putting these together, the analytic function

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

satisfies $\operatorname {Fib} (n)=F_{n}$ for all integers $n$.

Since $\operatorname {Fib} (z+2)=\operatorname {Fib} (z+1)+\operatorname {Fib} (z)$ 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,

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

Vector space

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.

Similar integer sequences

Fibonacci integer sequences

The 2-dimensional $\mathbb {Z}$-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:

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

where $\varphi$ 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 $(-\varphi )^{-1}$.

The sequence can be written in the form

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

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:

$L_{n}=\varphi ^{n}+(-\varphi )^{-n}.$

We have $L_{1}=1$ and $L_{2}=3$. The properties include:

${\begin{aligned}\varphi ^{n}&=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{\!n}={\frac {L(n)+F(n){\sqrt {5}}}{2}},\\L(n)&=F(n-1)+F(n+1).\end{aligned}}$

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.

See also Fibonacci integer sequences modulo n.

Lucas sequences

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

${\begin{aligned}U(0)&=0\\U(1)&=1\\U(n+2)&=PU(n+1)-QU(n),\end{aligned}}$

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 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, ... (sequence A006190 in the OEIS)

The 4-Fibonacci sequence is

0, 1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020, 1762289, 7465176, 31622993, 133957148, 567451585, 2403763488, ... (sequence A001076 in the OEIS)

The 5-Fibonacci sequence is

0, 1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275, 13741001, 71351280, 370497401, 1923838285, 9989688826, ... (sequence A052918 in the OEIS)

The 6-Fibonacci sequence is

0, 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, 12484830, 76934989, 474094764, 2921503573, 18003116202, ... (sequence A005668 in the OEIS)

The n-Fibonacci constant is the ratio toward which adjacent $n$-Fibonacci numbers tend; it is also called the nth metallic mean, and it is the only positive root of $x^{2}-nx-1=0$. For example, the case of $n=1$ is ${\frac {1+{\sqrt {5}}}{2}}$, or the golden ratio, and the case of $n=2$ is $1+{\sqrt {2}}$, or the silver ratio. Generally, the case of $n$ is ${\frac {n+{\sqrt {n^{2}+4}}}{2}}$.

Generally, $U(n)$ can be called (P,−Q)-Fibonacci sequence, and V(n) can be called (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, ... (sequence A001045 in the OEIS)

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, ... (sequence A006130 in the OEIS)

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, ... (sequence A002605 in the OEIS)

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, ... (sequence A030195 in the OEIS)

Fibonacci numbers of higher order

A Fibonacci sequence of order 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.

Tribonacci numbers

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, … (sequence A000073 in the OEIS)

The series was first described formally by Agronomof in 1914, 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. The term tribonacci was suggested by Feinberg in 1963.

The tribonacci constant

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

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.

A geometric construction of the Tribonacci constant (AC), with compass and marked ruler, according to the method described by Xerardo Neira.

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

$\xi ={\frac {{\sqrt[{3}]{17+3{\sqrt {33}}}}-{\sqrt[{3}]{-17+3{\sqrt {33}}}}-1}{3}}={\frac {3}{1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}}\approx 0.543689012.$ (sequence A192918 in the OEIS)

The tribonacci numbers are also given by

$T(n)=\left\lfloor 3b\,{\frac {\left({\frac {1}{3}}\left(a_{+}+a_{-}+1\right)\right)^{n}}{b^{2}-2b+4}}\right\rceil$

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

${\begin{aligned}a_{\pm }&={\sqrt[{3}]{19\pm 3{\sqrt {33}}}}\,,\\b&={\sqrt[{3}]{586+102{\sqrt {33}}}}\,.\end{aligned}}$

Tetranacci numbers

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, … (sequence A000078 in the OEIS)

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 (sequence A086088 in the OEIS), and also satisfies the equation $x+x^{-4}=2$.

The tetranacci constant can be expressed in terms of radicals by the following expression:

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

where,

$u={\frac {11}{12}}-{\frac {1}{3}}{\sqrt[{3}]{\frac {65+3{\sqrt {1689}}}{2}}}+{\frac {1}{3}}{\sqrt[{3}]{\frac {-65+3{\sqrt {1689}}}{2}}}$

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

Higher orders

Pentanacci, hexanacci, heptanacci, octanacci and enneanacci numbers have been computed. The pentanacci numbers are:

0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, … (sequence A001591 in the OEIS)

Hexanacci numbers:

0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, … (sequence A001592 in the OEIS)

Heptanacci numbers:

0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, … (sequence A122189 in the OEIS)

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, ... (sequence A079262 in the OEIS)

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, ... (sequence A104144 in the OEIS)

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

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

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

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

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

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

The root $r$ is in the interval $2(1-2^{-n})<r<2$. 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 $3^{-n}<|r|<1$.

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

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

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

The kth element of the n-nacci sequence is given by

$F_{k}^{(n)}=\left\lfloor {\frac {r^{k-1}(r-1)}{(n+1)r-2n}}\right\rceil \!,$

where $\lfloor \cdot \rceil$ 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 ${\frac {1}{2^{m}}}F_{m+2}^{(n)}$.

Fibonacci word

Main article: Fibonacci word

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

$F_{n}:=F(n):={\begin{cases}{\text{b}}&n=0;\\{\text{a}}&n=1;\\F(n-1)+F(n-2)&n>1.\\\end{cases}}$

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

b, a, ab, aba, abaab, abaababa, abaababaabaab, … (sequence A106750 in the OEIS)

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.

Convolved Fibonacci sequences

A convolved Fibonacci sequence is obtained applying a convolution operation to the Fibonacci sequence one or more times. Specifically, define

$F_{n}^{(0)}=F_{n}$

and

$F_{n}^{(r+1)}=\sum _{i=0}^{n}F_{i}F_{n-i}^{(r)}$

The first few sequences are

$r=1$: 0, 0, 1, 2, 5, 10, 20, 38, 71, … (sequence A001629 in the OEIS).

$r=2$: 0, 0, 0, 1, 3, 9, 22, 51, 111, … (sequence A001628 in the OEIS).

$r=3$: 0, 0, 0, 0, 1, 4, 14, 40, 105, … (sequence A001872 in the OEIS).

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

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

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

$F_{n}^{(r)}=r!F_{n}^{(r)}(1)$

where $F_{n}^{(r)}(x)$ is the $r$th derivative of $F_{n}(x)$. Equivalently, $F_{n}^{(r)}$ is the coefficient of $(x-1)^{r}$ when $F_{x}(x)$ is expanded in powers of $(x-1)$.

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

$F_{n}^{(1)}={\frac {nL_{n}-F_{n}}{5}}$

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 $F_{n}^{(1)}$ are the row sums of Hosoya's triangle.

As with Fibonacci numbers, there are several combinatorial interpretations of these sequences. For example $F_{n}^{(1)}$ 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 $F_{4}^{(1)}=5$ and 2 can be written 0 + 1 + 1, 0 + 2, 1 + 0 + 1, 1 + 1 + 0, 2 + 0.

Other generalizations

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 (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, … (sequence A007629 in the OEIS)

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 S. For example, if S is the Lucas sequence 2, 1, 3, 4, 7, 11, ..., then we obtain

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

N-generated Fibonacci sequence

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

$N=2^{a_{1}}\cdot 3^{a_{2}}\cdot 5^{a_{3}}\cdot 7^{a_{4}}\cdot 11^{a_{5}}\cdot 13^{a_{6}}\cdot \ldots \cdot p_{r}^{a_{r}},$

where p_r is the rth prime, then we define

$F_{N}(n)=a_{1}F_{N}(n-1)+a_{2}F_{N}(n-2)+a_{3}F_{N}(n-3)+a_{4}F_{N}(n-4)+a_{5}F_{N}(n-5)+...$

If $n=r-1$, then $F_{N}(n)=1$, and if $n<r-1$, then $F_{N}(n)=0$.

Sequence	N	OEIS sequence
Fibonacci sequence	6	A000045
Pell sequence	12	A000129
Jacobsthal sequence	18	A001045
Narayana's cows sequence	10	A000930
Padovan sequence	15	A000931
Third-order Pell sequence	20	A008998
Tribonacci sequence	30	A000073
Tetranacci sequence	210	A000288

Semi-Fibonacci sequence

The semi-Fibonacci sequence (sequence A030067 in the OEIS) 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 $a(2n)=a(n)$, $n\geq 1$. The bissection 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, ... (sequence A030068 in the OEIS)

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

External links

• "Tribonacci number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]