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A000045
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Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
(Formerly M0692 N0256)
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2468
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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; internal format)
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OFFSET
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COMMENTS
| Also called Lam{\'e}'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 (deutsch(AT)duke.poly.edu), Jun 18 2001
F(n) = number of compositions of n+1 with no part equal to 1 [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 (amarnath_murthy(AT)yahoo.com), 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 (rlahaye(AT)new.rr.com), 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 (malesevic(AT)kiklop.etf.bg.ac.yu), 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 (ck6(AT)evansville.edu), 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 (ck6(AT)evansville.edu), Jun 22 2004
F(n) = number of Catalan paths between the lines y = 0 and y = 3 from (0,0) to (n, GCD(n,2)). - Clark Kimberling (ck6(AT)evansville.edu), 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 (ck6(AT)evansville.edu), 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 (Gerald.McGarvey(AT)comcast.net), 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 (ck6(AT)evansville.edu), Nov 28 2004. [Correct by Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 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 (ck6(AT)evansville.edu), 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 (jsondow(AT)alumni.princeton.edu), Dec 19 2004
Lengths of successive words (starting with a) under the substitution: {a -> ab, b -> a} - J. F. J. Laros (jlaros(AT)liacs.nl), 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 (blekraj(AT)yahoo.com), Apr 29 2005
a(n) = Sum(abs(A108299(n, k)): 0 <= k <= n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005
a(n) = A001222(A000304(n)).
Fib(n+2)=sum(k=0..n, binomial(floor((n+k)/2),k) ), row sums of A04685 4. - Paul Barry (pbarry(AT)wit.ie), Mar 11 2003
Number of order ideals of the "zig-zag" poset. See vol. 1, ch. 3, prob. 23 of Stanley. - Mitch Harris (Harris.Mitchell (AT) mgh.harvard.edu), 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 (joshua.zucker(AT)stanfordalumni.org), 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 (Harris.Mitchell(AT)mgh.harvard.edu), 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 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 A. Davey (B.Davey(AT)latrobe.edu.au), Aug 25 2006
Number of Kekule structures in polyphenanthrenes. See the paper by Lukovits and Janezic for details. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), 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
Comment from Larry Gerstein (gerstein(AT)math.ucsb.edu), Mar 30 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.
Suppose psi=ln(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 (Hieronymus.Fischer(AT)gmx.de), Apr 18 2007
Inverse: floor(log_phi(sqr(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 (Hieronymus.Fischer(AT)gmx.de), 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 (jvospost3(AT)gmail.com), 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 (qntmpkt(AT)yahoo.com), Dec 15 2007
The sequence of first differences, fib(n+1)-fib(n), is essentailly 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 (azarian(AT)evansville.edu), May 21 2008
Equals row sums of triangle A144152. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]
Contribution from Cino Hilliard (hillcino368(AT)gmail.com), Sep 15 2008: (Start)
Except for the initial term, the numerator of the convergents to the recursion x
= 1/(x+1). (End)
Contribution from Ross Drewe (rd(AT)labyrinth.net.au), Oct 05 2008: (Start)
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. (End)
F(n) = PRODUCT_{k=1, (n-1)/2} (1 + 4*Cos^2 k*pi/n); where terms = roots to the Fibonacci product polynomials, A152063. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2008]
Fp == 5^((p-1)/2) mod p, p = prime; [Schroeder, p. 90]. [From Gary W. Adamson & Alexander Povolotsky (qntmpkt(AT)yahoo.com), Feb 21 2009]
(Ln)^2 - 5*(Fn)^2 = 4*(-1)^n. Example: 11^2 - 5*5 = -4. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 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. [From 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. [From 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. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 08 2009]
a(n+2)=A083662(A131577(n)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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. [From Henry Bottomley (se16(AT)btinternet.com), 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). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 11 2010]
(F(n-1)+F(n+1))^2 - 5F(n-2)*F(n+2) = 9*(-1)^n. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Mar 31 2010]
Pinter and Ziegler 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. [From Jonathan Vos Post (jvospost3(AT)gmail.com), May 22 2010]
F(n+1) = number of paths of length n starting at initial node on the path graph P_4. [From Johannes W. Meijer (meijgia(AT)hotmail.com), 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, 31.5.2010) [From Vaclav Kotesovec (kotesovec(AT)chello.cz), Jun 07 2010]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 16 2010: (Start)
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... (End)
Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 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-nint(k*phi)| > |Fib(n)*phi-nint(Fib(n)*phi)| holds.
Fib(n)*phi-nint(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.[From Gary Detlefs (gdetlefs(at)aol.com) Nov 27 2010]
F(n+k)^2-F(n)^2= F(k)*F(2n+k), for even k [From Gary Detlefs (gdetlefs(at)aol.com) Dec 04 2010]
F(n+k)^2+F(n)^2 = F(k)*F(2n+k), for odd k [From Gary Detlefs (gdetlefs(at)aol.com) 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. [From Joseph P. Shoulak (joepshoulak(at)me.com) Jan 13 2012]
For n > 0: a(n) = length of n-th row in Wythoff array A003603. [Reinhard Zumkeller, Jan 26 2012]
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REFERENCES
| Mohammad K. Azarian, The Generating Function for the Fibonacci Sequence, Missouri Journal of Mathematical Sciences, Vol. 2, No. 2, Spring 1990, pp. 78-79. Zentralblatt MATH, Zbl 1097.11516.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.
P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 70.
R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventues in Applied Mathematics, Princeton Univ. Press, 1999. See p. 84.
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178; http://www.combinatorics.org/Volume_18/PDF/v18i1p178.pdf.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 4.
Majorie Bicknell and Verner E Hoggatt, Fibonacci's Problem Book, Fibonacci Association, San Jose, Calif., 1974.
S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
N. D. Cahill and D. A. Narayan. "Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants". Fibonacci Quarterly, 42(3):216-221, 2004. V.E. Hoggatt and C.T. Long. "Divisibility Properties of Generalized Fibonacci Polynomials"; Fibonacci Quaterly, 12:113-130, 1974 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2008]
Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.
B. A. Davey and H. A. Priestley, Introduction to Lattices and Order (2nd edition), CUP, 2002. (See Exercise 1.15.)
B. Davis, 'The law of first digits' in 'Science Today'(subsequently renamed '2001')March 1980 pp. 55, Times of India, Mumbai.
Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
Reinhardt Euler, The Fibonacci Number of a Grid Graph and a New Class of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.6.
Helaman and Claire Ferguson, "Celebrating Mathematics in Stone and Bronze", Notices of the American Mathematical Society, 57 (2010), 840-850. See page 844. [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Aug 10 2010]
Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.
R. P. Grimaldi, Compositions without the summand 1, Proceedings Thirty-second Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 2001). Congr. Numer. 152 (2001), 33-43.
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H. Halberstam and K. F. Roth, Sequences, Oxord, 1966; see Appendix.
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V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
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P. W. Kasteleyn, The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice, Physica, 27(1961), 1209-1225. [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]
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Lukovits et al., Nanotubes: Number of Kekule structures and aromaticity, J. Chem. Inf. Comput. Sci, (2003), vol. 43, 609-614. See eq. 2 on page 610.
I. Lukovits and D. Janezic, "Enumeration of conjugated circuits in nanotubes", J. Chem. Inf. Comput. Sci., vol. 44, 410-414 (2004). See Table 1 second column.
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A. Sapounakis, I. Tasoulas and P. Tsikouras, On the Dominance Partial Ordering of Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.5.
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S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.
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D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 61-7, Penguin Books 1987.
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LINKS
| N. J. A. Sloane, The first 2000 Fibonacci numbers: Table of n, F(n) for n = 0..2000
Amazing Mathematical Object Factory, Information on the Fibonacci sequences
M. Anderson et al., The Fibonacci Series
Matt Anderson, Jeffrey Frazier and Kris Popendorf, The Fibonacci series: the successor formula
Matt Anderson, Jeffrey Frazier and Kris Popendorf, The Fibonacci series: the section index
P. G. Anderson, Fibonacci Facts
Joerg Arndt, Fxtbook
Author?, Fibonacci numbers from MathTV
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
H. Bottomley and N. J. A. Sloane, Illustration of initial terms: the Fibonacci tree
Brantacan, Fibonacci Numbers
J. Britton & B. V. Eeckhout, Fibonacci Interactive
N. D. Cahill, J. R. D'Errico, J. P. Spence, Complex factorizations of the Fibonacci and Lucas numbers, Fib. Quart. 41 (2003) 13
C. K. Caldwell, The Prime Glossary, Fibonacci number
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. S. Croot, Notes on Linear Recurrence Sequences
C. Dement, Posting to Math Forum.
R. M. Dickau, Fibonacci numbers
R. Doroslovacki, Binary sequences without 011...110 (k-1 1's) for fixed k, Mat. Vesnik 46 (1994), no. 3-4, 93-98.
Enthios LLC, Fibonacci Primer
G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255. MR1990179
Philipp Fahr and Claus Michael Ringel, Categorification of the Fibonacci Numbers Using Representations of Quivers
Helaman and Claire Ferguson, Celebrating Mathematics in Stone and Bronze
D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux,
I. Galkin, "Fibonacci Numbers Spelled Out"
Dale Gerdemann, Video of Fibonacci tree
Dale Gerdemann, Video of Fibonacci tree(s) (another version)
Dale Gerdemann, Golden Ratio Base Contains Zeckendorf and Negative Indexed Bunder Forms
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R. Knott, Fibonacci numbers and the golden section
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Author?, Fibonacci Pigeons [From Sarah Spolaor, Sep 30 2010]
Index entries for "core" sequences
Index entries for related partition-counting sequences
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (1,1)
Index to divisibility sequences
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FORMULA
| G.f.: x/(1-x-x^2).
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))
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 (kristmikl(AT)freemail.hu), Nov 13 2003
Kurmang. Aziz. Rashid (Kurmang.Rashid(AT)Btopenworld.com), 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: 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 (Gerald.McGarvey(AT)comcast.net), Jul 21 2004
|2*Fib(n) - 9*Fib(n+1)| = 4*A000032(n) + A000032(n+1). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug 13 2004
For x > Phi, Sum n=0..inf F(n)/x^n = x/(x^2 - x - 1) - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), 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 (jsondow(AT)alumni.princeton.edu), Dec 19 2004
a(n-1)=sum(k=0, n, (-1)^k*binomial(n-ceil(k/2), floor(k/2))) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 05 2005
F(n+1)=sum{k=0..n, binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2}; - Paul Barry (pbarry(AT)wit.ie), 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 (Hieronymus.Fischer(AT)gmx.de), Feb 07 2006
a(n)=(b(n+1)+b(n-1))/n where {b(n)} is the sequence A001629 - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), 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
Comments from Miklos Kristof (kristmikl(AT)freemail.hu), 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)
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 (Hieronymus.Fischer(AT)gmx.de), 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 (Hieronymus.Fischer(AT)gmx.de), 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 (qntmpkt(AT)yahoo.com), 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 (bpcrtz(AT)free.fr), 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 (qntmpkt(AT)yahoo.com), 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 (Ramsey2879(AT)msn.com), 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) + ..... ]. - Andre F. Labossiere (boronali(AT)laposte.net), Nov 24 2004
a(n+1)=Sum_{k, 0<=k<=n} A109466(n,k)*(-1)^(n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 26 2008]
Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), 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) [From 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)| [From Henry Bottomley (se16(AT)btinternet.com), 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. [From Milan R. Janjic (agnus(AT)blic.net), 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). [From Johannes W. Meijer (meijgia(AT)hotmail.com), 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. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jun 24 2010, Jun 27 2010]
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). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jul 01 2010]
a(n)=2*a(n-2)+a(n-3), n>2 [Gary Detlefs(gdetlefs(AT)aol.com), Sep 08 2010]
a(2^n)=Prod{i=0,...,n-1} A000032(2^i). [From Vladimir Shevelev(shevelev(AT)bgu.ac.il), 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 [From Gary Detlefs (gdetlefs(at)aol.com) Dec 03 2010]
F(2*n) = F(n+2)^2 - F(n+1)^2 - 2*F(n)^2 [From Rick Forberg (rrforberg(at)yahoo.com) 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). [From Joseph P. Shoulak (joepshoulak(at)me.com) Feb 05 2012]
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EXAMPLE
| Contribution from Cino Hilliard (hillcino368(AT)gmail.com), Sep 15 2008: (Start)
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 the 2nd to 6-th entries in the sequence. (End)
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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 (zerinvarylajos(AT)yahoo.com), Apr 04 2008
spec := [ B, {B=Sequence(Set(Z, card>1))}, unlabeled ]: seq(combstruct[count](spec, size=n), n=1..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2008
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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) [From 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*)
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PROG
| (AXIOM) [fibonacci(n) for n in 0..50]
(MAGMA) [Fibonacci(n): n in [0..38]];
(PARI) a(n)=fibonacci(n)
(PARI) a(n)=imag(quadgen(5)^n)
# Python program from 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:
# To see which functions are available type: sloane.A[tab]
# All builtin SAGE programs are called the same way:
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.
Contribution from Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Sep 29 2009: (Start)
(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] -} (End)
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CROSSREFS
| Cf. A039834 (signed Fibonacci numbers).
Cf. 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.
First row of array A103323. Second row of array A099390.
Row 2 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
a(n) = A094718(4, n). a(n) = A101220(0, j, n).
Cf. A000032 (F(n+1)+F(n-1)). Cf. A060441.
a(k) = A090888(0, k+1) = A118654(0, k+1) = A118654(1, k-1) = A109754(0, k) = A109754(1, k-1), for k > 0.
Cf. A144152, A152063, A114690.
Cf. A003893 (Fibonacci(n) mod 10).
Sequence in context: A132636 A152163 A039834 * A020695 A132916 A177194
Adjacent sequences: A000042 A000043 A000044 * A000046 A000047 A000048
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KEYWORD
| core,nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 30 2009
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