Inmathematics, theFibonacci numbers form asequence definedrecursively by:

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.

Using⁠${\displaystyle F_{n-2}=F_n-F_{n-1}}$⁠, one can extend the Fibonacci numbers to negativeintegers. So we get:

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

and⁠${\displaystyle F_{-n}=(-1{)}^{n+1}{F}_n}$⁠.^[1]

See alsoNegafibonacci coding.

There are a number of possible generalizations of the Fibonacci numbers which include thereal numbers (and sometimes thecomplex numbers) in their domain. These each involve thegolden ratioφ, and are based onBinet's formula

⁠${\displaystyle F_n=\frac{{\phi }^n-(-\phi {)}^{-n}}{\sqrt{5}}}$⁠.

Theanalytic function

${\displaystyle \mathrm{Fe}(x)=\frac{{\phi }^x-{\phi }^{-x}}{\sqrt{5}}}$

has the property that${\displaystyle \mathrm{Fe}(n)=F_n}$ foreven integers⁠${\displaystyle n}$⁠.^[2] Similarly, the analytic function:

${\displaystyle \mathrm{Fo}(x)=\frac{{\phi }^x+{\phi }^{-x}}{\sqrt{5}}}$

satisfies${\displaystyle \mathrm{Fo}(n)=F_n}$ forodd integers⁠${\displaystyle n}$⁠.

Finally, putting these together, the analytic function

${\displaystyle \mathrm{Fib}(x)=\frac{{\phi }^x-\cos (x\pi ){\phi }^{-x}}{\sqrt{5}}}$

satisfies${\displaystyle \mathrm{Fib}(n)=F_n}$ for all integers⁠${\displaystyle n}$⁠.^[3]

Since${\displaystyle \mathrm{Fib}(z+2)=\mathrm{Fib}(z+1)+\mathrm{Fib}(z)}$ for all complex numbers⁠${\displaystyle z}$⁠, 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,

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

However, this extension is by no means unique. For example, either

${\displaystyle \mathrm{Fib}(x)=\frac{{\phi }^x-\cos (kx\pi ){\phi }^{-x}}{\sqrt{5}}}$ or

${\displaystyle \mathrm{Fib}(x)=\frac{{\phi }^x-\exp (ikx\pi ){\phi }^{-x}}{\sqrt{5}}}$

for any odd integerk is an extension of the Fibonacci number sequence to the entire complex plane, as is anylinear combination of them for which the coefficients sum to 1.

The termFibonacci sequence is also applied more generally to anyfunction${\displaystyle g}$ from the integers to afield for which⁠${\displaystyle g(n+2)=g(n)+g(n+1)}$⁠. These functions are precisely those of the form⁠${\displaystyle 1}$⁠, so the Fibonacci sequences form avector space with the functions${\displaystyle F(n)}$ and${\displaystyle F(n-1)}$ as abasis.

More generally, the range of${\displaystyle g}$ may be taken to be anyabelian group (regarded as aZ-module). Then the Fibonacci sequences form a 2-dimensionalZ-module in the same way.

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

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

where${\displaystyle \phi }$ is the golden ratio.

The ratio between two consecutive elementsconverges 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${\displaystyle (-\phi {)}^{-1}}$⁠.

The sequence can be written in the form

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

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

⁠${\displaystyle L_n={\phi }^n+(-\phi {)}^{-n}}$⁠.

We have${\displaystyle L_1=1}$ and⁠${\displaystyle L_2=3}$⁠. The properties include:

Every nontrivial Fibonacci integer sequence appears (possibly after a shift by a finite number of positions) as one of the rows of theWythoff array. The Fibonacci sequence itself is the first row, and a shift of the Lucas sequence is the second row.^[4]

See alsoFibonacci integer sequences modulon.

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

where the normal Fibonacci sequence is the special case of${\displaystyle P=1}$ and⁠${\displaystyle Q=-1}$⁠. Another kind of Lucas sequence begins with⁠${\displaystyle V(0)=2}$⁠,⁠${\displaystyle V(1)=P}$⁠. Such sequences have applications innumber theory andprimality proving.

When⁠${\displaystyle Q=-1}$⁠, this sequence is calledP-Fibonacci sequence, for example,Pell sequence is also called2-Fibonacci sequence.

The3-Fibonacci sequence is

0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, 467280, 1543321, 5097243, 16835050, 55602393, 183642229, 606529080, ... (sequenceA006190 in theOEIS)

The4-Fibonacci sequence is

0, 1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020, 1762289, 7465176, 31622993, 133957148, 567451585, 2403763488, ... (sequenceA001076 in theOEIS)

The5-Fibonacci sequence is

0, 1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275, 13741001, 71351280, 370497401, 1923838285, 9989688826, ... (sequenceA052918 in theOEIS)

The6-Fibonacci sequence is

0, 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, 12484830, 76934989, 474094764, 2921503573, 18003116202, ... (sequenceA005668 in theOEIS)

Then-Fibonacci constant is the ratio toward which adjacent${\displaystyle n}$-Fibonacci numbers tend; it is also called thenthmetallic mean, and it is the only positiveroot of⁠${\displaystyle x^2-nx-1=0}$⁠. For example, the case of${\displaystyle n=1}$ is⁠${\displaystyle \frac{1+\sqrt{5}}{2}}$⁠, or thegolden ratio, and the case of${\displaystyle n=2}$ is⁠${\displaystyle 1+\sqrt{2}}$⁠, or thesilver ratio. Generally, the case of${\displaystyle n}$ is⁠${\displaystyle \frac{n+\sqrt{n^2+4}}{2}}$⁠.^{[citation needed]}

Generally,${\displaystyle U(n)}$ can be called(P,−Q)-Fibonacci sequence, andV(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, ... (sequenceA001045 in theOEIS)

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, ... (sequenceA006130 in theOEIS)

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, ... (sequenceA002605 in theOEIS)

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, ... (sequenceA030195 in theOEIS)

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

These sequences, their limiting ratios, and the limit of these limiting ratios, were investigated byMark Barr in 1913.^[5]

Thetribonacci 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, … (sequenceA000073 in theOEIS)

The sequence was first described formally by Agronomof^[6] in 1914,^[7] but its first unintentional use is in theOrigin of Species byCharles R. Darwin. In the example of illustrating the growth of elephant population, he relied on the calculations made by his son,George H. Darwin.^[8] The termtribonacci was suggested by Feinberg in 1963.^[9]

Thetribonacci constant

${\displaystyle \frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3}=\frac{1+4\cosh \left(\frac{1}{3}{\cosh }^{-1}\left(2+\frac{3}{8}\right)\right)}{3}}$

is the ratio toward which adjacent tribonacci numbers tend. It is the unique real root of the polynomial⁠${\displaystyle x^3-x^2-x-1=0}$⁠, approximately1.839286755214161... (sequenceA058265 in theOEIS), and also satisfies the equation⁠${\displaystyle x+x^{-3}=2}$⁠. It is important in the study of thesnub cube.

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

${\displaystyle \xi =\frac{\sqrt[3]{17+3\sqrt{33}}-\sqrt[3]{-17+3\sqrt{33}}-1}{3}=\frac{3}{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}},}$ approximately0.543689012692076... (sequenceA192918 in theOEIS)

The tribonacci numbers are also given by^[10]

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

where${\displaystyle \lfloor \cdot \rceil }$ denotes thenearest integer function and

Thetetranacci 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, … (sequenceA000078 in theOEIS)

Thetetranacci constant is the ratio toward which adjacent tetranacci numbers tend. It is the unique positive real root of the polynomial⁠${\displaystyle x^4-x^3-x^2-x-1=0}$⁠, approximately1.927561975482925... (sequenceA086088 in theOEIS), and also satisfies the equation⁠${\displaystyle x+x^{-4}=2}$⁠.

The tetranacci constant can be expressed in terms ofradicals by the following expression:^[11]

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

where,

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

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

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, … (sequenceA001591 in theOEIS)

Thepentanacci constant is the ratio toward which adjacent pentanacci numbers tend. It is the unique real root of the polynomial⁠${\displaystyle x^5-x^4-x^3-x^2-x-1=0}$⁠, approximately1.965948236645485... (sequenceA103814 in theOEIS), and also satisfies the equation⁠${\displaystyle x+x^{-5}=2}$⁠.

Hexanacci numbers:

0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, … (sequenceA001592 in theOEIS)

Thehexanacci constant is the ratio toward which adjacent hexanacci numbers tend. It is the unique positive real root of the polynomial⁠${\displaystyle x^6-x^5-x^4-x^3-x^2-x-1=0}$⁠, approximately1.983582843424326... (sequence A118427 in theOEIS), and also satisfies the equation⁠${\displaystyle x+x^{-6}=2}$⁠.

Heptanacci numbers:

0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, … (sequenceA122189 in theOEIS)

Theheptanacci constant is the ratio toward which adjacent heptanacci numbers tend. It is the unique real root of the polynomial⁠${\displaystyle x^7-x^6-x^5-x^4-x^3-x^2-x-1=0}$⁠, approximately1.991964196605035... (sequenceA118428 in theOEIS), and also satisfies the equation⁠${\displaystyle x+x^{-7}=2}$⁠.

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, ... (sequenceA079262 in theOEIS)

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, ... (sequenceA104144 in theOEIS)

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 thepowers of two.

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

The limit of the ratio for any${\displaystyle n\ge 2}$ is the unique positive root of the characteristic equation^[11]

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

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

The above formulas for the ratio hold even for${\displaystyle n}$-nacci series generated from arbitrary starting numbers. The ratio approaches 2 in the limit that${\displaystyle n}$ increases to infinity.

The root${\displaystyle x}$ is in theinterval⁠⁠. The negative root of the characteristic equation is in the interval (−1, 0) when${\displaystyle n}$ is even. This root and each complex root of the characteristic equation hasmodulus⁠⁠.^[11]

A series for the positive root${\displaystyle x}$ for any${\displaystyle n>0}$ is^[11]

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

There is no solution of the characteristic equation in terms of radicals when5 ≤n ≤ 11.^[11]

Thekth element of then-nacci sequence is given by

${\displaystyle F_k^{(n)}=\lfloor \frac{x^{k-1}(x-1)}{(n+1)x-2n}\rceil ,}$

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

Acoin-tossing problem is related to the${\displaystyle n}$-nacci sequence. The probability that no${\displaystyle n}$ consecutive tails will occur in${\displaystyle m}$ tosses of an idealized coin is⁠${\displaystyle \frac{1}{2^m}{F}_{m+2}^{(n)}}$⁠.^[12]

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

where${\displaystyle +}$ denotes theconcatenation of two strings. The sequence of Fibonacci strings starts:

b, a, ab, aba, abaab, abaababa, abaababaabaab, … (sequenceA106750 in theOEIS)

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 theworst case in some computeralgorithms.

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

Aconvolved Fibonacci sequence is obtained applying aconvolution operation to the Fibonacci sequence one or more times. Specifically, define^[13]

${\displaystyle F_n^{(0)}=F_n}$

and

${\displaystyle F_n^{(r+1)}=\sum _{i=0}^n{F}_i{F}_{n-i}^{(r)}}$

The first few sequences are

${\displaystyle r=1}$: 0, 0, 1, 2, 5, 10, 20, 38, 71, … (sequenceA001629 in theOEIS).

${\displaystyle r=2}$: 0, 0, 0, 1, 3, 9, 22, 51, 111, … (sequenceA001628 in theOEIS).

${\displaystyle r=3}$: 0, 0, 0, 0, 1, 4, 14, 40, 105, … (sequenceA001872 in theOEIS).

The sequences can be calculated using the recurrence

${\displaystyle F_{n+1}^{(r+1)}=F_n^{(r+1)}+F_{n-1}^{(r+1)}+F_n^{(r)}}$

Thegenerating function of the${\displaystyle r}$th convolution is

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

The sequences are related to the sequence ofFibonacci polynomials by the relation

${\displaystyle F_n^{(r)}=r!F_n^{(r)}(1)}$

where${\displaystyle F_n^{(r)}(x)}$ is the${\displaystyle r}$thderivative of⁠${\displaystyle F_n(x)}$⁠. Equivalently,${\displaystyle F_n^{(r)}}$ is thecoefficient of${\displaystyle (x-1{)}^r}$ when${\displaystyle F_x(x)}$ is expanded in powers of⁠${\displaystyle (x-1)}$⁠.

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

${\displaystyle F_n^{(1)}=\frac{n{L}_n-F_n}{5}}$

and follows the recurrence

⁠${\displaystyle F_{n+1}^{(1)}=2F_n^{(1)}+F_{n-1}^{(1)}-2F_{n-2}^{(1)}-F_{n-3}^{(1)}}$⁠.

Similar expressions can be found for${\displaystyle r>1}$ with increasing complexity as${\displaystyle r}$ increases. The numbers${\displaystyle F_n^{(1)}}$ are the row sums ofHosoya's triangle.

As with Fibonacci numbers, there are several combinatorial interpretations of these sequences. For example${\displaystyle F_n^{(1)}}$ is the number of ways${\displaystyle n-2}$ can be written as an ordered sum involving only 0, 1, and 2 with 0 used exactly once. In particular${\displaystyle F_4^{(1)}=5}$ and 2 can be written0 + 1 + 1,0 + 2,1 + 0 + 1,1 + 1 + 0,2 + 0.^[14]

TheFibonacci polynomials are another generalization of Fibonacci numbers.

ThePadovan sequence is generated by the recurrence⁠${\displaystyle P(n)=P(n-2)+P(n-3)}$⁠.

TheNarayana's cows sequence is generated by the recurrence⁠${\displaystyle N(k)=N(k-1)+N(k-3)}$⁠.

Arandom Fibonacci sequence can be defined by tossing a coin for each position${\displaystyle n}$ of the sequence and taking${\displaystyle F(n)=F(n-1)+F(n-2)}$ if it lands heads and${\displaystyle F(n)=F(n-1)-F(n-2)}$ if it lands tails. Work by Furstenberg and Kesten guarantees that this sequencealmost surelygrows exponentially at a constant rate: the constant is independent of the coin tosses and was computed in 1999 byDivakar Viswanath. It is now known asViswanath's constant.

Arepfigit, orKeith 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, … (sequenceA007629 in theOEIS)

Since the set of sequences satisfying the relation${\displaystyle S(n)=S(n-1)+S(n-2)}$ is closed under termwise addition and under termwise multiplication by a constant, it can be viewed as avector 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${\displaystyle (S(0),S(1))}$⁠, the Fibonacci sequence${\displaystyle F(n)=(0,1)}$ and the shifted Fibonacci sequence${\displaystyle F(n-1)=(1,0)}$ are seen to form a canonical basis for this space, yielding the identity:

${\displaystyle S(n)=S(0)F(n-1)+S(1)F(n)}$

for all such sequencesS. For example, ifS is the Lucas sequence2, 1, 3, 4, 7, 11, ..., then we obtain

⁠${\displaystyle L(n)=2F(n-1)+F(n)}$⁠.

We can define theN-generated Fibonacci sequence (whereN is a positiverational number): if

${\displaystyle N=2^{a_1}\cdot 3^{a_2}\cdot 5^{a_3}\cdot 7^{a_4}\cdot {11}^{a_5}\cdot {13}^{a_6}\cdot \dots \cdot p_r^{a_r},}$

wherep_r is therth prime, then we define

${\displaystyle F_N(n)=a_1{F}_N(n-1)+a_2{F}_N(n-2)+a_3{F}_N(n-3)+a_4{F}_N(n-4)+a_5{F}_N(n-5)+\dots }$

If⁠${\displaystyle n=r-1}$⁠, then⁠${\displaystyle F_N(n)=1}$⁠, and if⁠⁠, then⁠${\displaystyle F_N(n)=0}$⁠.^{[citation needed]}

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

Thesemi-Fibonacci sequence (sequenceA030067 in theOEIS) is defined via the same recursion for odd-indexed terms${\displaystyle a(2n+1)=a(2n)+a(2n-1)}$ and⁠${\displaystyle a(1)=1}$⁠, but for even indices⁠${\displaystyle a(2n)=a(n)}$⁠,⁠${\displaystyle n\ge 1}$⁠. The bisectionA030068 of odd-indexed terms${\displaystyle s(n)=a(2n-1)}$ therefore verifies${\displaystyle s(n+1)=s(n)+a(n)}$ and isstrictly increasing. It yields the set of thesemi-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, ... (sequenceA030068 in theOEIS)

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

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

• Fibonacci numbers

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