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A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
(Formerly M0692 N0256)
3246
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Also called Lamé's sequence.

F(n+2) = number of binary sequences of length n that have no consecutive 0's.

F(n+2) = number of subsets of {1,2,...,n} that contain no consecutive integers.

F(n+1) = number of tilings of a 2 X n rectangle by 2 X 1 dominoes.

F(n+1) = number of matchings in a path graph on n vertices: F(5)=5 because the matchings of the path graph on the vertices A, B, C, D are the empty set, {AB}, {BC}, {CD} and {AB, CD}. - Emeric Deutsch, Jun 18 2001

F(n) = number of compositions of n+1 with no part equal to 1. [Cayley, Grimaldi]

Positive terms are the solutions to z = 2*x*y^4 + (x^2)*y^3 - 2*(x^3)*y^2 - y^5 - (x^4)*y + 2*y for x,y >= 0 (Ribenboim, page 193). When x=F(n), y=F(n + 1) and z>0 then z=F(n + 1).

For Fibonacci search see Knuth, Vol. 3; Horowitz and Sahni; etc.

F(n) is the diagonal sum of the entries in Pascal's triangle at 45 degrees slope. - Amarnath Murthy, Dec 29 2001

F(n+1) is the number of perfect matchings in ladder graph L_n = P_2 X P_n. - Sharon Sela (sharonsela(AT)hotmail.com), May 19 2002

F(n+1) = number of (3412,132)-, (3412,213)- and (3412,321)-avoiding involutions in S_n.

This is also the Horadam sequence (0,1,1,1). - Ross La Haye, Aug 18 2003

An INVERT transform of A019590. INVERT([1,1,2,3,5,8,...]) gives A000129. INVERT([1,2,3,5,8,13,21,...]) gives A028859. - Antti Karttunen, Dec 12 2003

Number of meaningful differential operations of the k-th order on the space R^3. - Branko Malesevic, Mar 02 2004

F(n)=number of compositions of n-1 with no part greater than 2. Example: F(4)=3 because we have 3 = 1+1+1=1+2=2+1.

F(n) = number of compositions of n into odd parts; e.g., F(6) counts 1+1+1+1+1+1, 1+1+1+3, 1+1+3+1, 1+3+1+1, 1+5, 3+1+1+1, 3+3, 5+1. - Clark Kimberling, Jun 22 2004

F(n) = number of binary words of length n beginning with 0 and having all runlengths odd; e.g., F(6) counts 010101, 010111, 010001, 011101, 011111, 000101, 000111, 000001. - Clark Kimberling, Jun 22 2004

The number of sequences (s(0),s(1),...s(n)) such that 0<s(i)<5, |s(i)-s(i-1)|=1 and s(0)=1 is F(n+1); e.g., F(5+1) = 8 corresponds to 121212, 121232, 121234, 123212, 123232, 123234, 123432, 123434. - Clark Kimberling, Jun 22 2004. [Corrected by Neven Juric, Jan 09 2009]

Likewise F(6+1) = 13 corresponds to these thirteen sequences with seven numbers: 1212121, 1212123, 1212321, 1212323, 1212343, 1232121, 1232123, 1232321, 1232323, 1232343, 1234321, 1234323, 1234343. - Neven Juric, Jan 09 2008

A relationship between F(n) and the Mandelbrot set is discussed in the link "Le nombre d'or dans l'ensemble de Mandelbrot" (in French). - Gerald McGarvey, Sep 19 2004

For n>0, the continued fraction for F(2n-1)*Phi=[F(2n);L(2n-1),L(2n-1),L(2n-1),...] and the continued fraction for F(2n)*Phi=[F(2n+1)-1;1,L(2n)-2,1,L(2n)-2,...]. Also true: F(2n)*Phi=[F(2n+1);-L(2n),L(2n),-L(2n),L(2n),...] where L(i) is the i-th Lucas number (A000204).... - Clark Kimberling, Nov 28 2004. [Corrected by Hieronymus Fischer, Oct 20 2010]

F(n) = number of permutations p of 1,2,3,...,n such that |k-p(k)|<=1 for k=1,2,...,n. (For <=2 and <=3, see A002524 and A002526.) - Clark Kimberling, Nov 28 2004

The ratios F(n+1)/F(n) for n>0 are the convergents to the simple continued fraction expansion of the golden section. - Jonathan Sondow, Dec 19 2004

Lengths of successive words (starting with a) under the substitution: {a -> ab, b -> a}. - Jeroen F.J. Laros, Jan 22 2005

The Fibonacci sequence, like any additive sequence, naturally tends to be geometric with common ratio not a rational power of 10; consequently, for a sufficiently large number of terms, Benford's law of first significant digit (i.e., first digit 1 =< d =< 9 occurring with probability log_10(d+1) - log_10(d)) holds. - Lekraj Beedassy, Apr 29 2005

a(n) = Sum(abs(A108299(n, k)): 0 <= k <= n). - Reinhard Zumkeller, Jun 01 2005

a(n) = A001222(A000304(n)).

Fib(n+2)=sum(k=0..n, binomial(floor((n+k)/2),k)), row sums of A046854. - Paul Barry, Mar 11 2003

Number of order ideals of the "zig-zag" poset. See vol. 1, ch. 3, prob. 23 of Stanley. - Mitch Harris, Dec 27 2005

F(n+1)/F(n) is also the Farey fraction sequence (see A097545 for explanation) for the golden ratio, which is the only number whose Farey fractions and continued fractions are the same. - Joshua Zucker, May 08 2006

a(n+2) is the number of paths through 2 plates of glass with n reflections (reflections occurring at plate/plate or plate/air interfaces). Cf. A006356-A006359. - Mitch Harris, Jul 06 2006

F(n+1) equals the number of downsets (i.e., decreasing subsets) of an n-element fence, i.e., an ordered set of height 1 on {1,2,...,n} with 1 > 2 < 3 > 4 < ... n and no other comparabilities. Alternatively, F(n+1) equals the number of subsets A of {1,2,...,n} with the property that, if an odd k is in A, then the adjacent elements of {1,2,...,n} belong to A, i.e., both k - 1 and k + 1 are in A (provided they are in {1,2,...,n}). - Brian Davey, Aug 25 2006

Number of Kekule structures in polyphenanthrenes. See the paper by Lukovits and Janezic for details. - Parthasarathy Nambi, Aug 22 2006

Inverse: With phi = (sqrt(5) + 1)/2, round(log_phi(sqrt((sqrt(5) a(n) + sqrt(5 a(n)^2 - 4))(sqrt(5) a(n) + sqrt(5 a(n)^2 + 4)))/2)) = n for n >= 3, obtained by rounding the arithmetic mean of the inverses given in A001519 and A001906. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007

A result of Jacobi from 1848 states that every symmetric matrix over a p.i.d. is congruent to a triple-diagonal matrix. Consider the maximal number T(n) of summands in the determinant of an n X n triple-diagonal matrix. This is the same as the number of summands in such a determinant in which the main-, sub- and super-diagonal elements are all nonzero. By expanding on the first row we see that the sequence of T(n)'s is the Fibonacci sequence without the initial stammer on the 1's. - Larry Gerstein (gerstein(AT)math.ucsb.edu), Mar 30 2007

Suppose psi=log(phi). We get the representation F(n)=(2/sqrt(5))*sinh(n*psi) if n is even; F(n)=(2/sqrt(5))*cosh(n*psi) if n is odd. There is a similar representation for Lucas numbers (A000032). Many Fibonacci formulas now easily follow from appropriate sinh- and cosh-formulas. For example: the de Moivre theorem (cosh(x)+sinh(x))^m=cosh(mx)+sinh(mx) produces L(n)^2+5F(n)^2=2L(2n) and L(n)F(n)=F(2n) (setting x=n*psi and m=2). - Hieronymus Fischer, Apr 18 2007

Inverse: floor(log_phi(sqrt(5)*Fib(n))+0.5)=n, for n>1. Also for n>0, floor(1/2*log_phi(5*Fib(n)*Fib(n+1)))=n. Extension valid for integer n, except n=0,-1: floor(1/2*sign(Fib(n)*Fib(n+1))*log_phi|5*Fib(n)*Fib(n+1)|)=n (where sign(x) = sign of x). - Hieronymus Fischer, May 02 2007

F(n+2) = The number of Khalimsky-continuous functions with a two-point codomain. - Shiva Samieinia (shiva(AT)math.su.se), Oct 04 2007

From Kauffman and Lopes, Proposition 8.2, p. 21: "The sequence of the determinants of the Fibonacci sequence of rational knots is the Fibonacci sequence (of numbers)." - Jonathan Vos Post, Oct 26 2007

This is a_1(n) in the Doroslovacki reference.

Let phi = 1.6180339...; then phi^n = (1/phi)*a(n) + a(n+1). Example: phi^4 = 6.8541019...= (.6180339...)*3 + 5. Also phi = 1/1 + 1/2 + 1/(2*5) + 1/(5*13) + 1/(13*34) + 1/(34*89),... - Gary W. Adamson, Dec 15 2007

The sequence of first differences, fib(n+1)-fib(n), is essentially the same sequence: 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... - Colm Mulcahy, Mar 03 2008

a(n)= the number of different ways to run up a staircase with n steps, taking steps of odd sizes where the order is relevant and there is no other restriction on the number or the size of each step taken. - Mohammad K. Azarian, May 21 2008

Equals row sums of triangle A144152. - Gary W. Adamson, Sep 12 2008

Except for the initial term, the numerator of the convergents to the recursion x = 1/(x+1). - Cino Hilliard, Sep 15 2008

F(n) is the number of possible binary sequences of length n that obey the sequential construction rule: if last symbol is 0, add the complement (1); else add 0 or 1. Here 0,1 are metasymbols for any 2-valued symbol set. This rule has obvious similarities to JFJ Laros's rule, but is based on addition rather than substitution and creates a tree rather than a single sequence. - Ross Drewe, Oct 05 2008

F(n) = PRODUCT_{k=1, (n-1)/2} (1 + 4*cos^2 k*Pi/n); where terms = roots to the Fibonacci product polynomials, A152063. - Gary W. Adamson, Nov 22 2008

Fp == 5^((p-1)/2) mod p, p = prime; [Schroeder, p. 90]. - Gary W. Adamson & Alexander R. Povolotsky, Feb 21 2009

(Ln)^2 - 5*(Fn)^2 = 4*(-1)^n. Example: 11^2 - 5*5 = -4. - Gary W. Adamson, Mar 11 2009

Output of Kasteleyn's formula for the number of perfect matchings of an m x n grid specializes to the Fibonacci sequence for m=2. - Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009

(Fib(n),Fib(n+4)) satisfies the Diophantine equation: X^2 + Y^2 - 7XY = 9*(-1)^n. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 06 2009

(Fib(n),Fib(n+2)) satisfies the Diophantine equation: X^2 + Y^2 - 3XY = (-1)^n. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 08 2009

a(n+2)=A083662(A131577(n)). - Reinhard Zumkeller, Sep 26 2009

Difference between of number of closed walks of length n+1 from a node on a pentagon and number of walks of length n+1 between two adjacent nodes on a pentagon. - Henry Bottomley, Feb 10 2010

F(n+1) = number of Motzkin paths of length n having exactly one weak ascent. A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1) and H=(1,0) steps and never going below the x-axis. A weak ascent in a Motzkin path is a maximal sequence of consecutive U and H steps. Example: a(5)=5 because we have (HHHH), (HHU)D, (HUH)D, (UHH)D, and (UU)DD (the unique weak ascent is shown between parentheses; see A114690). - Emeric Deutsch, Mar 11 2010

(F(n-1)+F(n+1))^2 - 5F(n-2)*F(n+2) = 9*(-1)^n. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Mar 31 2010

From the Pinter and Ziegler reference's abstract: authors "show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic progressions is also given." - Jonathan Vos Post, May 22 2010

F(n+1) = number of paths of length n starting at initial node on the path graph P_4. - Johannes W. Meijer, May 27 2010

F(k) = Number of cyclotomic polynomials in denominator of generating function for number of ways to place k nonattacking queens on an n X n board. - Vaclav Kotesovec, Jun 07 2010

As n-> inf., (a(n)/a(n-1) - a(n-1)/a(n)) tends to 1.0. Example: a(12)/a(11) - a(11)/a(12) = 144/89 - 89/144 = .99992197.... - Gary W. Adamson, Jul 16 2010

From Hieronymus Fischer, Oct 20 2010: (Start)

Fibonacci numbers are those numbers m such that m*phi is closer to an integer than k*phi for all k, 1<=k<m. More formally: a(0)=0, a(1)=1, a(2)=1, a(n+1)=minimal m>a(n) such that m*phi is closer to an integer than a(n)*phi.

For all numbers 1<=k<Fib(n), the inequality |k*phi-round(k*phi)| > |Fib(n)*phi-round(Fib(n)*phi)| holds.

Fib(n)*phi - round(Fib(n)*phi) = -((-phi)^(-n)), for n>1.

fract(0.5+Fib(n)*phi) = 0.5 -(-phi)^(-n), for n>1.

fract(Fib(n)*phi) = (1/2)*(1+(-1)^n)-(-phi)^(-n), n>1.

Inverse: n = -log_phi |0.5-fract(0.5+Fib(n)*phi)|. (End)

F(A001177(n)*k) mod n = 0, for any integer k. - Gary Detlefs, Nov 27 2010

F(n+k)^2-F(n)^2 = F(k)*F(2n+k), for even k. - Gary Detlefs, Dec 04 2010

F(n+k)^2+F(n)^2 = F(k)*F(2n+k), for odd k. - Gary Detlefs, Dec 04 2010

"Even the Fibonacci sequence - 1,1,2,3,5,8,13 - follows Benford's law." See Pickover.

F(n) = round(phi* F(n-1)) for n>1. - Joseph P. Shoulak, Jan 13 2012

For n > 0: a(n) = length of n-th row in Wythoff array A003603. - Reinhard Zumkeller, Jan 26 2012

From Bridget Tenner, Feb 22, 2012: (Start)

The number of free permutations of [n].

The number of permutations of [n] for which s_k in supp(w) implies s_{k+-1} not in supp(w).

The number of permutations of [n] in which every decomposition into length(w) reflections is actually composed of simple reflections. (End)

The sequence F(n+1)^(1/n) is increasing. The sequence F(n+2)^(1/n) is decreasing. - Thomas Ordowski, Apr 19 2012

Two conjectures: For n > 1, F(n+2)^2 mod F(n+1)^2 = F(n)*F(n+1) - (-1)^n. For n > 0, (F(2n) + F(2n+2))^2 = F(4n+3) + sum_{k = 2..2n}F(2k). - Alex Ratushnyak, May 06 2012

From Ravi Kumar Davala, Jan 30 2014, (Start)

Proof of Ratushnyak's first conjecture: For n > 1, F(n+2)^2 - F(n)*F(n+1)  + (-1)^n = 2F(n+1)^2.

Consider: F(n+2)^2 - F(n)*F(n+1) - 2F(n+1)^2

         = F(n+2)^2 - F(n+1)^2 - F(n+1)^2 - F(n)*F(n+1)

         =(F(n+2) + F(n+1))*(F(n+2) - F(n+1)) - F(n+1)*(F(n+1) + F(n))

         = F(n+3)*F(n) - F(n+1)*F(n+2) = -(-1)^n.

Proof of second conjecture: L(n) stands for Lucas number sequence from A000032.

consider the fact

    L(2n+1)^2 = L(4n+2) - 2

   (F(2n) + F(2n+2))^2 = F(4n+1) + F(4n+3) - 2

   (F(2n) + F(2n+2))^2 = sum{k = 2..2n, F(2k)} + F(4n+3). (end)

The relationship: INVERT transform of (1,1,0,0,0,...) = (1, 2, 3, 5, 8,...), while the INVERT transform of (1,0,1,0,1,0,1,...) = (1, 1, 2, 3, 5, 8,...) is equivalent to: The numbers of compositions using parts 1 and 2 is equivalent to the numbers of compositions using parts == 1 mod 2 (i.e., the odd integers). Generally, the numbers of compositions using parts 1 and k is equivalent to the numbers of compositions of (n+1) using parts 1 mod k. Cf. A000930 for k = 3 and A003269 for k = 4. Example: for k = 2, n = 4 we have the compositions (22; 211, 121; 112; 1111) = 5; but using parts 1 and 3 we have for n = 5: (311, 131, 113, 11111, 5) = 5. - Gary W. Adamson, Jul 05 2012

The sequence F(n) is the binomial transformation of the alternating sequence (-1)^(n-1)*F(n), whereas the sequence F(n+1) is the binomial transformation of the alternating sequence (-1)^n*F(n-1). Both of these facts follow easily from the equalities a(n;1)=F(n+1) and b(n;1)=F(n) where a(n;d) and b(n;d) are so-called "delta-Fibonacci" numbers as defined in comments to A014445 (see also Witula et al.'s papers). - Roman Witula, Jul 24 2012

F(n) is the number of different (n-1)-digit binary numbers such that all substrings of length > 1 have at least one digit equal to 1. Example: for n = 5 there are 8 binary numbers with n - 1 = 4 digits (1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111), only the F(n) = 5 numbers 1010, 1011, 1101, 1110 and 1111 have the desired property. - Hieronymus Fischer, Nov 30 2012

For positive n, F(n+1) equals the determinant of the n X n tridiagonal matrix with 1's along the main diagonal, i's along the superdiagonal and along the subdiagonal where i = sqrt(-1). Example: Det([1,i,0,0; i,1,i,0; 0,i,1,i; 0,0,i,1]) = F(4+1) = 5. - Philippe Deléham, Feb 24 2013

For n>=1, number of compositions of n where there is a drop between every second pair of parts, starting with the first and second part; see example. Also, a(n+1) is the number of compositions where there is a drop between every second pair of parts, starting with the second and third part; see example. - Joerg Arndt, May 21 2013

Central terms of triangles in A162741 and A208245, n > 0. - Reinhard Zumkeller, Jul 28 2013

For n>=4, F(n-1) is the number of simple permutations in the geometric grid class given in A226433. - Jay Pantone, Sep 08 2013

From Wolfdieter Lang, Oct 01 2013: (Start)

a(n) are the pentagon (not pentagonal) numbers because the algebraic degree 2 number rho(5) = 2*cos(pi/5) = phi (golden section), the length ratio diagonal/side in a pentagon, has minimal polynomial C(5,x) = x^2 - x - 1 (see A187360, n=5), hence rho(5)^n = a(n-1)*1 + a(n)*rho(5), n >= 0, in the power basis of the algebraic number field Q(rho(5)). One needs a(-1) = 1 here. See also the P. Steinbach reference under A049310. (End)

A010056(a(n)) = 1. - Reinhard Zumkeller, Oct 10 2013

Define F(-n) to be F(n) for n odd and -F(n) for n even. Then for all n and k, F(n+2k)^2 - F(n)^2 = F(n+k)*( F(n+3k) - F(n-k) ). - Charlie Marion, Dec 20 2013

( F(n), F(n+2k) ) satisfies the Diophantine equation: X^2 + Y^2 - L(2k)*X*Y = F(4k)^2*(-1)^n.  This generalizes Bouhamida’s comments dated Sep 06 2009 and Sep 08 2009. - Charlie Marion, Jan 07 2014

For any prime p there is an infinite periodic subsequence within F(n) divisible by p, that begins at index n = 0 with value 0, and its first nonzero term at n = A001602(i), and period k = A001602(i). Also see A236479. - Richard R. Forberg, Jan 26 2014

Range of row n of the circular Pascal array of order 5. - Shaun V. Ault, May 30 2014 (orig. Kicey-Klimko 2011, and observations by Glen Whitehead. More general work found in Ault-Kicey 2014.)

Nonnegative range of the quintic polynomial 2*y - y^5 + 2*x*y^4 + x^2*y^3 - 2*x^3*y^2 - x^4*y with x, y >= 0, see Jones 1975. - Charles R Greathouse IV, Jun 01 2014

The expression round(1/(F(k+1)/F(n) + F(k)/F(n+1))), for n > 0, yields a Fibonacci sequence with k-1 leading zeros (with rounding 0.5 to 0). - Richard R. Forberg, Aug 04 2014

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Index entries for "core" sequences

Index to divisibility sequences

Index entries for related partition-counting sequences

Index to sequences with linear recurrences with constant coefficients, signature (1,1).

Index entries for two-way infinite sequences

FORMULA

G.f.: x / (1 - x - x^2).

G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + x)/(1 + k*x). - Paul D. Hanna, Oct 26 2013

F(n) = ((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^n*sqrt(5)).

Alternatively, F(n) = ((1/2+sqrt(5)/2)^n-(1/2-sqrt(5)/2)^n)/sqrt(5).

F(n) = F(n-1) + F(n-2) = -(-1)^n F(-n).

F(n) = round(phi^n/sqrt(5)).

F(n+1) = Sum(0 <= j <= [n/2]; binomial(n-j, j)).

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - Michael Somos, Apr 07 2012

E.g.f.: (2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 30 2001

[0 1; 1 1]^n [0 1] = [F(n); F(n+1)]

x | F(n) ==> x | F(kn).

A sufficient condition for F(m) to be divisible by a prime p is (p - 1) divides m, if p == 1 or 4 (mod 5); (p + 1) divides m, if p == 2 or 3 (mod 5); or 5 divides m, if p = 5. (This is essentially Theorem 180 in Hardy and Wright.) - Fred W. Helenius (fredh(AT)ix.netcom.com), Jun 29 2001

a(n)=F(n) has the property: F(n)*F(m) + F(n+1)*F(m+1) = F(n+m+1). - Miklos Kristof, Nov 13 2003

Kurmang. Aziz. Rashid, Feb 21 2004, makes 4 conjectures and gives 3 theorems:

Conjecture 1: for n>=2 sqrt{F(2n+1)+F(2n+2)+F(2n+3)+F(2n+4)+2*(-1)^n}={F(2n+1)+2*(-1)^n}/F(n-1).

Conjecture 2: for n>=0, {F(n+2)* F(n+3)}-{F(n+1)* F(n+4)}+ (-1)^n = 0.

Conjecture 3: for n>=0, F(2n+1)^3 - F(2n+1)*[(2*A^2) -1] - [A + A^3]=0, where A = {F(2n+1)+sqrt{5*F(2n+1)^2 +4}}/2.

Conjecture 4: for x>=5, if x is a Fibonacci number >= 5 then g*x*[{x+sqrt{5*(x^2) +- 4}}/2]*[2x+{{x+sqrt{5*(x^2) +- 4}}/2}]*[2x+{{3x+3*sqrt {5*(x^2) +- 4}}/2}]^2+[2x+{{x+sqrt{5*(x^2) +- 4}}/2}] +- x*[2x+{{3x+3*sqrt{5*(x^2) +- 4}}/2}]^2 -x*[2x+{{x+sqrt{5*(x^2) +- 4}}/2}]*[x+{{x+sqrt{5*(x^2) +- 4}}/2}]* [2x+ {{3x+3*sqrt{5*(x^2) +- 4}}/2}]^2= 0, where g = {1 + sqrt 5 /2}.

Theorem 1: for n>=0, {F(n+3)^ 2 - F(n+1)^ 2}/F(n+2)={F(n+3)+ F(n+1)}.

Theorem 2: for n>=0, F(n+10) = 11* F(n+5) + F(n).

Theorem 3: for n>=6, F(n) = 4* F(n-3) + F(n-6).

Conjecture 2 of Rashid is actually a special case of the general law F(n)*F(m) + F(n+1)*F(m+1) = F(n+m+1) (take n <- n+1 and m <- -(n+4) in this law). - Harmel Nestra (harmel.nestra(AT)ut.ee), Apr 22 2005

Conjecture 2 of Rashid Kurmang simplified:  F(n)*F(n+3) = F(n+1)*F(n+2)-(-1)^n. Follows from d'Ocagne's identity: m=n+2. - Alex Ratushnyak, May 06 2012

Conjecture: for all c such that 2-Phi <= c < 2*(2-Phi) we have F(n) = floor(Phi*a(n-1)+c) for n > 2. - Gerald McGarvey, Jul 21 2004

|2*Fib(n) - 9*Fib(n+1)| = 4*A000032(n) + A000032(n+1). - Creighton Dement, Aug 13 2004

For x > Phi, Sum n=0..inf F(n)/x^n = x/(x^2 - x - 1) - Gerald McGarvey, Oct 27 2004

F(n+1) = exponent of the n-th term in the series f(x, 1) determined by the equation f(x, y) = xy + f(xy, x). - Jonathan Sondow, Dec 19 2004

a(n-1)=sum(k=0, n, (-1)^k*binomial(n-ceil(k/2), floor(k/2))). - Benoit Cloitre, May 05 2005

F(n+1)=sum{k=0..n, binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2}. - Paul Barry, Aug 28 2005

Fibonacci(n)=Product(1 + 4[cos(j*Pi/n)]^2, j=1..ceil(n/2)-1). [Bicknell and Hoggatt, pp. 47-48.] - Emeric Deutsch, Oct 15 2006

F(n)=2^-(n-1)*sum{k=0..floor((n-1)/2), binomial(n,2*k+1)*5^k}. - Hieronymus Fischer, Feb 07 2006

a(n)=(b(n+1)+b(n-1))/n where {b(n)} is the sequence A001629. - Sergio Falcon, Nov 22 2006

F(n*m) = Sum{k = 0..m, binomial(m,k)*F(n-1)^k*F(n)^(m-k)*F(m-k)}. The generating function of F(n*m) (n fixed, m = 0,1,2...) is G(x) = F(n)*x / ((1-F (n-1)*x)^2-F(n)*x*(1-F(n-1)*x)-( F(n)*x)^2). E.g., F(15) = 610 = F(5*3) = binomial(3,0)* F(4)^0*F(5)^3*F(3) + binomial(3,1)* F(4)^1*F(5)^2*F(2) + binomial(3,2)* F(4)^2*F(5)^1*F(1) + binomial(3,3)* F(4)^3*F(5)^0*F(0) = 1*1*125*2 + 3*3*25*1 + 3*9*5*1 + 1*27*1*0 = 250 + 225 + 135 + 0 = 610. - Miklos Kristof, Feb 12 2007

From Miklos Kristof, Mar 19 2007: (Start)

Let L(n)=A000032=Lucas numbers. Then:

For a>=b and odd b, F(a+b)+F(a-b)=L(a)*F(b).

For a>=b and even b, F(a+b)+F(a-b)=F(a)*L(b).

For a>=b and odd b, F(a+b)-F(a-b)=F(a)*L(b).

For a>=b and even b, F(a+b)-F(a-b)=L(a)*F(b).

F(n+m)+(-1)^m*F(n-m)=F(n)*L(m);

F(n+m)-(-1)^m*F(n-m)=L(n)*F(m);

F(n+m+k)+(-1)^k*F(n+m-k)+(-1)^m*(F(n-m+k)+(-1)^k*F(n-m-k))=F(n)*L(m)*L(k);

F(n+m+k)-(-1)^k*F(n+m-k)+(-1)^m*(F(n-m+k)-(-1)^k*F(n-m-k))=L(n)*L(m)*F(k);

F(n+m+k)+(-1)^k*F(n+m-k)-(-1)^m*(F(n-m+k)+(-1)^k*F(n-m-k))=L(n)*F(m)*L(k);

F(n+m+k)-(-1)^k*F(n+m-k)-(-1)^m*(F(n-m+k)-(-1)^k*F(n-m-k))=5*F(n)*F(m)*F(k). (End)

A corollary to Kristof 2007 is 2*F(a+b)=F(a)*L(b)+L(a)*F(b) - Graeme McRae, Apr 24 2014

For n>m, the sum of the 2m consecutive Fibonacci numbers F(n-m-1) thru F(n+m-2) is F(n)*L(m) if m is odd, and L(n)*F(m) if m is even (see the McRae link). - Graeme McRae, Apr 24 2014.

Fib(n)=b(n)+(p-1)*sum{1<k<n, floor(b(k)/p)*Fib(n-k+1)} where b(k) is the digital sum analogue of the Fibonacci recurrence, defined by b(k)=ds_p(b(k-1))+ds_p(b(k-2)), b(0)=0, b(1)=1, ds_p=digital sum base p. Example for base p=10: Fib(n)=A010077(n)+9*sum{1<k<n, A059995(A010077(k))*Fib(n-k+1)}. - Hieronymus Fischer, Jul 01 2007

Fib(n)=b(n)+p*sum{1<k<n, floor(b(k)/p)*Fib(n-k+1)} where b(k) is the digital product analogue of the Fibonacci recurrence, defined by b(k)=dp_p(b(k-1))+dp_p(b(k-2)), b(0)=0, b(1)=1, dp_p=digital product base p. Example for base p=10: Fib(n)=A074867(n)+10*sum{1<k<n, A059995(A074867(k))*Fib(n-k+1)}. - Hieronymus Fischer, Jul 01 2007

a(n) = denominator of continued fraction [1,1,1,...] (with n ones); e.g., 2/3 = continued fraction [1,1,1]; where barover[1] = [1,1,1...] = .6180339.... - Gary W. Adamson, Nov 29 2007

F(n + 3) = 2F(n + 2) - F(n), F(n + 4) = 3F(n + 2) - F(n), F(n + 8) = 7F(n + 4) - F(n), F(n + 12) = 18F(n + 6) - F(n). - Paul Curtz, Feb 01 2008

1 = 1/(1*2) + 1/(1*3) + 1/(2*5) + 1/(3*8) + 1/(5*13) + ... = 1/2 + 1/3 + 1/10 + 1/24 + 1/65 + 1/168 + ...; where A059929 = (0, 2, 3, 10, 24, 65, 168,...). - Gary W. Adamson, Mar 16 2008

a(2^n) = prod{i=0}^{n-2}B(i) where B(i) is A001566. Example 3*7*47 = Fib(16). - Kenneth J Ramsey, Apr 23 2008

F(n) = (1/(n-1)!) * [ n^(n-1) - { C(n-2,0) +4*C(n-2,1) +3*C(n-2,2) }*n^(n-2) + { 10*C(n-3,0) +49*C(n-3,1) +95*C(n-3,2) +83*C(n-3,3) +27*C(n-3,4) }*n^(n-3) - { 90*C(n-4,0) +740*C(n-4,1) +2415*C(n-4,2) +4110*C(n-4,3) +3890*C(n-4,4) +1950*C(n-4,5) +405*C(n-4,6) }*n^(n-4) + ... ]. - André F. Labossière, Nov 24 2004

a(n+1)=Sum_{k, 0<=k<=n} A109466(n,k)*(-1)^(n-k). [Philippe Deléham, Oct 26 2008]

Formula from Thomas Wieder, Feb 25 2009:

a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1}

delta(l_1,l_2,...,l_i,...,l_n)

where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i + l_(i+1) >= 2 for i=1..n-1

and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise.

2^n (\prod _{k=1}^n \sqrt[4]{\cos^2(k\pi/(n+1))+1/4})^2 (Kasteleyn's formula specialized). - Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009

a(n+1) =sum_{k=floor[n/2] mod 5} C(n,k) - sum_{k=floor[(n+5)/2] mod 5} C(n,k) =A173125(n)-A173126(n) =|A054877(n)-A052964(n-1)|. - Henry Bottomley, Feb 10 2010

If p[i]=modp(i,2) and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, May 02 2010

Limit(F(k+n)/F(k), k = infinity) = (L(n) + F(n)*sqrt(5))/2 with the Lucas numbers L(n)= A000032(n). - Johannes W. Meijer, May 27 2010

For n>=1, F(n)=round(log_2(2^{\phi*F(n-1)}+2^{\phi*F(n-2)})), where \phi is Golden ratio. - Vladimir Shevelev, Jun 24 2010, Jun 27 2010

For n>=1, a(n+1)=ceil(phi*a(n)), if n is even and a(n+1)=floor(phi*a(n)), if n is odd (phi=golden ratio). - Vladimir Shevelev, Jul 01 2010

a(n)=2*a(n-2)+a(n-3), n>2. - Gary Detlefs, Sep 08 2010

a(2^n)=Prod{i=0,...,n-1} A000032(2^i). - Vladimir Shevelev, Nov 28 2010

a(n)^2 - a(n-1)^2 = a(n+1)*a(n-2), see A121646.

a(n) = sqrt((-1)^k*(a(n+k)^2-a(k)*a(2n+k))), for any k. - Gary Detlefs, Dec 03 2010

F(2*n) = F(n+2)^2 - F(n+1)^2 - 2*F(n)^2. - Richard R. Forberg, Jun 04 2011

From Artur Jasinski, Nov 17 2011: (Start)

  (-1)^(n+1) = F(n)^2 + F(n)*F(1+n) - F(1+n)^2.

  F(n) = -F(n+2)(-2+(F(n+1))^4 + 2*(F(n+1)^3*F(n+2)) - (F(n+1)*F(n+2))^2 2*F(n+1)(F(n+2))^3 + (F(n+2))^4)-F(n+1). (End)

F(n) = 1 + sum_{x=1..n-2} F(x). - Joseph P. Shoulak, Feb 05 2012

F(n) = 4*F(n-2) - 2*F(n-3) - F(n-6). - Gary Detlefs, Apr 01 2012

F(n) = round(phi^(n+1)/(phi+2)). - Thomas Ordowski, Apr 20 2012

From Sergei N. Gladkovskii, Jun 03 2012: (Start)

G.f. A(x) = x/(1-x-x^2) = G(0)/sqrt(5) where G(k)= 1 -((-1)^k)*2^k/(a^k - b*x*a^k*2^k/(b*x*2^k - 2*((-1)^k)*c^k/G(k+1))) and a=3+sqrt(5), b=1+sqrt(5), c=3-sqrt(5); (continued fraction, 3rd kind, 3-step).

Let E(x) be the e.g.f., i.e.,

E(x) = 1*x+1/2*x^2+1/3*x^3+1/8*x^4+1/24*x^5+1/90*x^6+13/5040*x^7+... then

E(x) = G(0)/sqrt(5); G(k)= 1 -((-1)^k)*2^k/(a^k - b*x*a^k*2^k/(b*x*2^k -  2*((-1)^k)*(k+1)*c^k/G(k+1))), where a=3+sqrt(5), b=1+sqrt(5), c=3-sqrt(5); (continued fraction, 3rd kind, 3-step).

(End)

From Hieronymus Fischer, Nov 30 2012: (Start)

Fib(n) = 1 + sum_{j_1=1..n-2} 1 + sum_{j_1=1..n-2} sum_{j_2=1..j_1-2} 1 + sum_{j_1=1..n-2} sum_{j_2=1..j_1-2} sum_{j_3=1..j_2-2} 1 + ... + sum_{j_1=1..n-2} sum_{j_2=1..j_1-2} sum_{j_3=1..j_2-2} ... sum_{j_k=1..j_(k-1)-2} 1, where k = floor((n-1)/2).

Example: Fib(6) = 1 + sum_{j=1..4} 1 + sum_{j=1..4} sum_{k=1..(j-2)} 1 + 0 = 1 + (1 + 1 + 1 + 1) + (1 + (1 + 1)) = 8.

Fib(n) = sum_{j=0..k} S(j+1,n-2j), where k = floor((n-1)/2) and the S(j,n) are the n-th j-simplex sums: S(1,n) = 1 is the 1-simplex sum, S(2,n) = sum_{k=1..n} S(1,k) = 1+1+...+1 = n is the 2-simplex sum, S(3,n) = sum_{k=1..n} S(2,k) = 1+2+3+…+n is the 3-simplex sum (= triangular numbers = A000217), S(4,n) = sum_{k=1..n} S(3,k) = 1+3+6+...+n(n+1)/2 is the 4-simplex sum (= tetrahedral numbers = A000292) and so on.

Since S(j,n) = binomial(n-2+j,j-1), the formula above equals the well-known binomial formula, essentially.

(End)

G.f. A(x) = x / (1 - x / (1 - x / (1 + x))). - Michael Somos, Jan 04 2013

sum{n>=1}(-1)^(n-1)/(a(n)*a(n+1)) = 1/phi (phi=golden ratio). - Vladimir Shevelev, Feb 22 2013

From Vladimir Shevelev, Feb 24 2013: (Start)

(1) Expression a(n+1) via a(n): a(n+1) = (a(n) + sqrt(5*(a(n))^2 + 4*(-1)^n))/2;

(2) sum_{k=1,...,n} (-1)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);

(3) a(n)/a(n+1) = 1/phi + r(n), where |r(n)| < 1/(a(n+1)*a(n+2)). (End)

F(n+1) = F(n)/2 + sqrt((-1)^n + 5*F(n)^2/4), n>=0. F(n+1) = U_n(i/2)/i^n, (U:= Chebyshef 2nd kind). - R. W. Gosper, Mar 04 2013

G.f.: -Q(0) where Q(k) = 1 - (1+x)/(1 - x/(x - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013

G.f.: x-1-1/x + 1/x/Q(0), where Q(k) = 1 - (k+1)*x/(1 - x/(x - (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 23 2013

G.f.: x*G(0), where G(k)= 1 + x*(1+x)/(1 - x*(1+x)/(x*(1+x) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 08 2013

G.f.: x^2 - 1 + 2*x^2/(W(0)-2), where W(k) = 1 + 1/(1 - x*(k + x)/( x*(k+1 + x) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013

G.f.: Q(0) -1, where Q(k) = 1 + x^2 + (k+2)*x -x*(k+1 + x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 06 2013

Let b(n) = b(n-1) + b(n-2), with b(0) = 0, b(1) = phi. Then, for n>=2, F(n)= floor(b(n-1)) if n is even, F(n) = ceil(b(n-1)), if n is odd, with convergence. - Richard R. Forberg, Jan 19 2014

a(n)= sum(t1*g(1)+t2*g(2)+...+tn*g(n)=n, multinomial(t1+t2 +...+tn,t1,t2,...,tn), where g(k)=2*k-1. - Mircea Merca, Feb 27 2014

F(n) = round(sqrt(F(n-1)^2 + F(n)^2 + F(n+1)^2)/2), for n > 0. This rule appears to apply to any sequence of the form  a(n) = a(n-1) + a(n-2), for any two values of a(0) and a(1), if n is sufficiently large. - Richard R. Forberg, Jul 27 2014

F(n) = round(2/(1/F(n) + 1/F(n+1) + 1/F(n+2)), for n > 0.  This rule also appears to apply to any sequence of the form  a(n) = a(n-1) + a(n-2), for any two values of a(0) and a(1), if n is sufficiently large. - Richard R. Forberg, Aug 03 2014

F(n) = round(1/sum(j>=n+2, 1/F(j))). - Richard R. Forberg, Aug 14 2014

a(n) = hypergeometric([-n/2+1/2, -n/2+1], [-n+1], -4) for n>=2. - Peter Luschny, Sep 19 2014

EXAMPLE

For x = 0,1,2,3,4, x=1/(x+1) = 1, 1/2, 2/3, 3/5, 5/8. These fractions have numerators 1,1,2,3,5, which are the 2nd to 6th entries in the sequence. - Cino Hilliard, Sep 15 2008

From Joerg Arndt, May 21 2013: (Start)

There are a(7)=13 compositions of 7 where there is a drop between every second pair of parts, starting with the first and second part:

01:  [ 2 1 2 1 1 ]

02:  [ 2 1 3 1 ]

03:  [ 2 1 4 ]

04:  [ 3 1 2 1 ]

05:  [ 3 1 3 ]

06:  [ 3 2 2 ]

07:  [ 4 1 2 ]

08:  [ 4 2 1 ]

09:  [ 4 3 ]

10:  [ 5 1 1 ]

11:  [ 5 2 ]

12:  [ 6 1 ]

13:  [ 7 ]

There are abs(a(6+1))=13 compositions of 6 where there is no rise between every second pair of parts, starting with the second and third part:

01:  [ 1 2 1 2 ]

02:  [ 1 3 1 1 ]

03:  [ 1 3 2 ]

04:  [ 1 4 1 ]

05:  [ 1 5 ]

06:  [ 2 2 1 1 ]

07:  [ 2 3 1 ]

08:  [ 2 4 ]

09:  [ 3 2 1 ]

10:  [ 3 3 ]

11:  [ 4 2 ]

12:  [ 5 1 ]

13:  [ 6 ]

(End)

MAPLE

A000045 := proc(n) combinat[fibonacci](n); end;

ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 1)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..38); - Zerinvary Lajos, Apr 04 2008

spec := [ B, {B=Sequence(Set(Z, card>1))}, unlabeled ]: seq(combstruct[count](spec, size=n), n=1..39); - Zerinvary Lajos, Apr 04 2008

MATHEMATICA

Table[ Fibonacci[ k ], {k, 1, 50} ]

2^(n) Product[((Cos[Pi k/(n + 1)])^2 + (Cos[Pi 1/3])^2)^(1/4), {k, n}] Product[((Cos[Pi k/(n + 1)])^2 + (Cos[Pi 2/3])^2)^(1/4), {k, n}] (* Kasteleyn's formula specialized, Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009 *)

Table[Fibonacci[n]^5 - Fibonacci[1 + n] + 3 Fibonacci[n]^4 Fibonacci[1 + n] + Fibonacci[n]^3 Fibonacci[1 + n]^2 - 3 Fibonacci[n]^2 Fibonacci[1 + n]^3 -   Fibonacci[n] Fibonacci[1 + n]^4 + Fibonacci[1 + n]^5, {n, 1, 10}] (* Artur Jasinski, Nov 17 2011 *)

LinearRecurrence[{1, 1}, {0, 1}, 40] (* Harvey P. Dale, Aug 03 2014 *)

PROG

(Axiom) [fibonacci(n) for n in 0..50]

(MAGMA) [Fibonacci(n): n in [0..38]];

(Maxima) makelist(fib(n), n, 0, 100); /* Martin Ettl, Oct 21 2012 */

(PARI) {a(n) = fibonacci(n)};

(PARI) {a(n) = imag(quadgen(5)^n)};

(PARI) a(n)=my(phi=quadgen(5)); (phi^n-(-1/phi)^n)/(2*phi-1) \\ Charles R Greathouse IV, Jun 17 2012

(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, k+x +x*O(x^n))/prod(k=1, m, 1+k*x +x*O(x^n))), n)} \\ Paul D. Hanna, Oct 26 2013

(Python) # Jaap Spies, Jan 05 2007 (Change leading dots to blanks.)

def fib():

... """ Generates the Fibonacci numbers, starting with 0 """

... x, y = 0, 1

... while 1:

....... yield x

....... x, y = y, x+y

.

f = fib()

a = [f.next() for i in range(100)]

.

def A000045(n):

... """ Returns Fibonacci number with index n, offset 0, 4 """

... return a[n]

................

def A000045_list(N):

... """ Returns a list of the first n Fibonacci numbers """

... return a[:N]

.

(Sage) ## Demonstration program from Jaap Spies:

a = sloane.A000045; ## choose sequence

print a ## This returns the name of the sequence.

print a(38) ## This returns the 38-th number of the sequence.

print a.list(39) ## This returns a list of the first 39 numbers.

(Haskell)

-- Based on code from http://www.haskell.org/haskellwiki/The_Fibonacci_sequence

-- which also has other versions.

fib :: Int -> Integer

fib n = fibs !! n

.. where

.... fibs = 0 : 1 : zipWith (+) fibs (tail fibs)

{- Example of use: map fib [0..38] Gerald McGarvey, Sep 29 2009 -}

(Sage) [i for i in fibonacci_sequence(0, 40)] # Bruno Berselli, Jun 26 2014

CROSSREFS

Cf. A039834 (signed Fibonacci numbers), A000213, A000288, A000322, A000383, A060455, A030186, A039834, A020695, A020701, A071679, A099731, A100492, A094216, A094638, A000108, A101399, A101400, A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616, A059929, A144152, A152063, A114690, A003893, A000032, A060441, A000930, A003269.

First row of arrays A103323, A234357. Second row of arrays A099390, A048887, and A092921 (k-generalized Fibonacci numbers).

a(n) = A094718(4, n). a(n) = A101220(0, j, n).

a(k) = A090888(0, k+1) = A118654(0, k+1) = A118654(1, k-1) = A109754(0, k) = A109754(1, k-1), for k > 0.

Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074.

Cf. A001690 (complement).

Boustrophedon transforms: A000738, A000744.

Sequence in context: A152163 A039834 * A236191 A020695 A212804 A132916

Adjacent sequences:  A000042 A000043 A000044 * A000046 A000047 A000048

KEYWORD

core,nonn,nice,easy,hear

AUTHOR

N. J. A. Sloane, Apr 30 1991

STATUS

approved

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Last modified October 25 20:39 EDT 2014. Contains 248557 sequences.