%I A000045 M0692 N0256
%S A000045 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,
%T A000045 10946,17711,28657,46368,75025,121393,196418,317811,514229,832040,
%U A000045 1346269,2178309,3524578,5702887,9227465,14930352,24157817,39088169
%N A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2), F(0) = 0, F(1) = 1, F(2) =
1, ...
%C A000045 Also called Lam{\'e}'s sequence.
%C A000045 F(n+2) = number of binary sequences of length n that have no consecutive
0's.
%C A000045 F(n+2) = number of subsets of {1,2,...,n} that contain no consecutive
integers.
%C A000045 F(n+1) = number of tilings of a 2 X n rectangle by 2 X 1 dominoes.
%C A000045 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
%C A000045 F(n) = number of compositions of n+1 with no part equal to 1 [Grimaldi]
%C A000045 Positive terms are the solutions to z = 2xy^4 + (x^2)y^3 - 2(x^3)y^2
- y^5 - (x^4)y + 2y for x,y >= 0 (Ribenboim, page 193). When x=F(n),
y=F(n + 1) and z>0 then z=F(n + 1).
%C A000045 For Fibonacci search see Knuth, Vol. 3; Horowitz and Sahni; etc.
%C A000045 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
%C A000045 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
%C A000045 F(n+1) = number of (3412,132)-, (3412,213)- and (3412,321)-avoiding involutions
in S_n.
%C A000045 This is also the Horadam sequence (0,1,1,1). - Ross La Haye (rlahaye(AT)new.rr.com),
Aug 18 2003
%C A000045 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
%C A000045 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
%C A000045 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.
%C A000045 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
%C A000045 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
%C A000045 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
%C A000045 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]
%C A000045 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.
%C A000045 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
%C A000045 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);
L(2n)-2,L(2n)-2,L(2n)-2,...] where L(i) is the i-th Lucas number
(A000204). - Clark Kimberling (ck6(AT)evansville.edu), Nov 28 2004
%C A000045 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
%C A000045 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
%C A000045 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
%C A000045 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
%C A000045 a(n) = Sum(abs(A108299(n, k)): 0 <= k <= n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Jun 01 2005
%C A000045 a(n) = A001222(A000304(n)).
%C A000045 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
%C A000045 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
%C A000045 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
%C A000045 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
%C A000045 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
%C A000045 Number of Kekule structures in polyphenanthrenes. See the paper by Lukovits
and Janezic for details. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com),
Aug 22 2006
%C A000045 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
%C A000045 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.
%C A000045 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
%C A000045 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
%C A000045 F(n+2) = The number of Khalimsky-continuous functions with a two-point
codomain. - Shiva Samieinia (shiva(AT)math.su.se), Oct 04 2007
%C A000045 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
%C A000045 This is a_1(n) in the Doroslovacki reference.
%C A000045 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
%C A000045 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
%C A000045 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
%C A000045 Equals row sums of triangle A144152. [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Sep 12 2008]
%C A000045 Contribution from Cino Hilliard (hillcino368(AT)gmail.com), Sep 15 2008:
(Start)
%C A000045 Except for the initial term, the numerator of the convergents to the
recursion x
%C A000045 = 1/(x+1). (End)
%C A000045 Contribution from Ross Drewe (rd(AT)labyrinth.net.au), Oct 05 2008: (Start)
%C A000045 F(n) is the number of possible binary sequences of length n that obey
the
%C A000045 sequential construction rule: if last symbol is 0, add the complement
(1);
%C A000045 else add 0 or 1. Here 0,1 are metasymbols for any 2-valued symbol set.
This
%C A000045 rule has obvious similarities to JFJ Laros's rule, but is based on addition
%C A000045 rather than substitution and creates a tree rather than a single sequence.
(End)
%C A000045 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]
%C A000045 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]
%C A000045 (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]
%C A000045 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]
%C A000045 (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]
%C A000045 Number of units of a(n) belongs to a periodic sequence: 0, 1, 1, 2, 3,
5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3,
8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6,
9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1.We conclude that a(n) and a(n+60)
have the same number of units. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr),
Sep 05 2009]
%C A000045 (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]
%C A000045 a(n+2)=A083662(A131577(n)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Sep 26 2009]
%D A000045 Mohammad K. Azarian, The Generating Function for the Fibonacci Sequence,
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%D A000045 Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem
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2004, pp. 12-17.
%D A000045 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY,
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%D A000045 R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventues in
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%D A000045 Paul Barry, A Catalan Transform and Related Transformations on Integer
Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A000045 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal
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%D A000045 B. A. Davey and H. A. Priestley, Introduction to Lattices and Order (2nd
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%D A000045 B. Davis, 'The law of first digits' in 'Science Today'(subsequently renamed
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%D A000045 Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by
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%D A000045 Reinhardt Euler, The Fibonacci Number of a Grid Graph and a New Class
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%D A000045 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.
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%D A000045 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers.
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%D A000045 J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden,
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%D A000045 V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA,
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%D A000045 E. Horowitz and S. Sahni, Fundamentals of Data Structures, Computer Science
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%D A000045 C. W. Huegy and D. B. West, A Fibonacci tiling of the plane, Discrete
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%D A000045 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley
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%D A000045 Lukovits et al., Nanotubes: Number of Kekule structures and aromaticity,
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%D A000045 I. Lukovits and D. Janezic, "Enumeration of conjugated circuits in nanotubes",
J. Chem. Inf. Comput. Sci., vol. 44, 410-414 (2004). See Table 1
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%D A000045 B. Malesevic: Some combinatorial aspects of differential operation composition
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%D A000045 Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas
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%D A000045 Mark A. Shattuck and Carl G. Wagner, Periodicity and Parity Theorems
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%D A000045 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
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%D A000045 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
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%D A000045 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the
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%D A000045 S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood
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%D A000045 N. N. Vorobiev, Fibonacci Numbers, Birkhauser (Basel;Boston) 2002.
%D A000045 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers,
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Chem. Modell., Vol. 4, (2006), pp. 405-469.
%D A000045 A. S. Posamentier & I. Lehmann, The Fabulous Fibonacci Numbers, Prometheus
Books, Amherst, NY 2007.
%D A000045 Aimei Yu and Xuezheng Lv, "The Merrifield-Simmons indices and Hosoya
indices of trees with k pendant vertices.", J. Math. Chem., Vol.
41 (2007), pp. 33-43. See page 35. - from Parthasarathy Nambi (PachaNambi(AT)yahoo.com),
Sep 01 2008
%D A000045 Manfred R. Schroeder, "Number Theory in Science and Communication", 5-th
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%D A000045 Mark A. Shattuck, Tiling proofs of some formulas for the Pell numbers
of odd index, Integers, 9 (2009), 53-64.
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%H A000045 N. J. A. Sloane, <a href="b000045.txt">The first 2000 Fibonacci numbers:
Table of n, F(n) for n = 0..2000</a>
%H A000045 Amazing Mathematical Object Factory, <a href="http://www.schoolnet.ca/
vp-pv/amof/e_fiboI.htm">Information on the Fibonacci sequences</a>
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The Fibonacci Series</a>
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27890/theSeries4.html">The Fibonacci series: the successor formula</
a>
%H A000045 Matt Anderson, Jeffrey Frazier and Kris Popendorf, <a href="http://library.thinkquest.org/
27890/theSeries.html">The Fibonacci series: the section index</a>
%H A000045 P. G. Anderson, <a href="http://www.cs.rit.edu/~pga/Fibo/fact_sheet.html">
Fibonacci Facts</a>
%H A000045 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A000045 Author?, <a href="http://www.youtube.com/watch?v=_NmSEEhtc1U">Fibonacci
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Fibonacci Numbers</a>
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fibonacci/jbfibapplet.htm">Fibonacci Interactive</a>
%H A000045 C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=FibonacciNumber">
Fibonacci Numbers</a>
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page.php/FibonacciNumber.html">Fibonacci number</a>
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Sequences realized by oligomorphic permutation groups</a>, J. Integ.
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%H A000045 C. Conner, <a href="http://www.geocities.com/cyd-conner/page1.html">Fibonacci</
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%H A000045 E. S. Croot, <a href="http://www.math.gatech.edu/~ecroot/recurrence_notes2.pdf">
Notes on Linear Recurrence Sequences</a>
%H A000045 C. Dement, <a href="http://mathforum.org/discuss/sci.math/t/622432">Posting
to Math Forum</a>.
%H A000045 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/fibboard.html">
Fibonacci numbers</a>
%H A000045 R. Doroslovacki, <a href="http://www.emis.de/journals/MV/9434/mv943407.ps">
Binary sequences without 011...110 (k-1 1's) for fixed k</a>, Mat.
Vesnik 46 (1994), no. 3-4, 93-98.
%H A000045 Enthios LLC, <a href="http://www.enthios.com/FibonacciPrimer.htm">Fibonacci
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%H A000045 G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http:/
/www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences and
Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002),
Article 02.2.3
%H A000045 D. Foata and G.-N. Han, <a href="http://www-irma.u-strasbg.fr/~foata/
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%H A000045 I. Galkin, <a href="http://ulcar.uml.edu/~iag/CS/Fibonacci.html">"Fibonacci
Numbers Spelled Out"</a>
%H A000045 L. Goldsmith, <a href="http://people.bath.ac.uk/ma2lag/fibonaccinumbers.html">
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%H A000045 A. P. Hillman & G. L. Alexanderson, Algebra Through Problem Solving,
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%H A000045 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=9">
Encyclopedia of Combinatorial Structures 9</a>
%H A000045 Tina Hill Janzen, <a href="http://www.youtube.com/watch?v=2nAycC7sGVI">
Fibonacci sequence / Golden scale</a> [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com),
Aug 02 2009]
%H A000045 R. Javonovic, <a href="http://milan.milanovic.org/math/english/function/
function.html">Fibonacci Function Calculator</a>
%H A000045 R. Javonovic, <a href="http://milan.milanovic.org/math/english/pdf/Fibonacci.pdf">
The relations between the Fibonacci and the Lucas numbers</a>
%H A000045 R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviFibo">
First 70 Fibonacci numbers</a>
%H A000045 S. Kak, <a href="http://uk.arXiv.org/abs/physics/0411195">The Golden
Mean and the Physics of Aesthetics</a>
%H A000045 Louis H. Kauffman and Pedro Lopes, <a href="http://arXiv.org/pdf/0710.3765">
Graded forests and rational knots</a>, Oct 19, 2007.
%H A000045 B. Kelly, <a href="http://home.att.net/~blair.kelly/mathematics/fibonacci/
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%H A000045 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
RecursiveSequences.html">Recursive Sequences</a>
%H A000045 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Matrix Transformations of Integer Sequences</a>, J. Integer Seqs.,
Vol. 6, 2003.
%H A000045 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
">Fibonacci numbers and the golden section</a>
%H A000045 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
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%H A000045 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
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%H A000045 A. Krowne, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
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%H A000045 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
Sobalian Coefficients</a>.
%H A000045 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
index.html">Miscellaneous</a>.
%H A000045 Hendrik Lenstra, <a href="http://math.berkeley.edu/~hwl/papers/fibo.pdf">
Profinite Fibonacci Numbers</a>
%H A000045 M. A. Lerma, <a href="http://www.math.northwestern.edu/~mlerma/problem_solving/
results/recurrences.pdf">Recurrence Relations</a>
%H A000045 D. Litchfield, D. Goldenheim and C. H. Dietrich, <a href="http://scientium.com/
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%H A000045 B. Malesevic, <a href="http://matematika.etf.bg.ac.yu/publikacije/pub/
P09(98)/P09_06.ZIP">Some combinatorial aspects of differential operation
composition on the space R^n </a>.
%H A000045 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/
~merlini/tiling.ps">Strip tiling and regular grammars</a>, Theoret.
Computer Sci. 242, 1-2 (2000) 109-124.
%H A000045 D. Merrill, <a href="http://pw1.netcom.com/~merrills/fibphi.html">The
Fib-Phi Link Page</a>
%H A000045 Jean-Christophe Michel, <a href="http://framy.free.fr/fibonacci%20dans%20mandelbrot.htm">
Le nombre d'or dans l'ensemble de Mandelbrot</a> (in French, 'The
golden number in the Mandelbrot set')
%H A000045 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/
matha108.htm">Factorizations of many number sequences</a>
%H A000045 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/
matha109.htm">Factorizations of many number sequences</a>
%H A000045 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/
matha110.htm">Factorizations of many number sequences</a>
%H A000045 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/
matha111.htm">Factorizations of many number sequences</a>
%H A000045 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/
matha112.htm">Factorizations of many number sequences</a>
%H A000045 P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved
Fibonacci numbers</a>
%H A000045 Newton's Institute, <a href="http://www.newton.cam.ac.uk/wmy2kposters/
january">Posters in the London Underground</a>
%H A000045 J. Patterson, <a href="http://www.bath.ac.uk/~ma1jmp/link.html">The Fibonacci
Sequence</a>
%H A000045 Ivars Peterson, <a href="http://www.sciencenews.org/articles/20060603/
mathtrek.asp">Fibonacci's Missing Flowers</a>.
%H A000045 S. Plouffe, Project Gutenberg, <a href="http://ibiblio.org/pub/docs/books/
gutenberg/etext01/fbncc10.txt">The First 1001 Fibonacci Numbers</
a>
%H A000045 S. Plouffe, <a href="http://www.lacim.uqam.ca/~plouffe/OEIS/A000045">
Fibonacci numbers</a> [Contains the first 754965 terms]
%H A000045 S. Rabinowitz, <a href="http://www.mathpropress.com/stan/bibliography/
algorithmicFib.pdf">Algorithmic Manipulation of Fibonacci Identities</
a> (1996). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov
06 2008]
%H A000045 Marc Renault, <a href="http://www.math.temple.edu/~renault/fibonacci/
thesis.html">Properties of the Fibonacci sequence under various moduli</
a>, MSc Thesis, Wake Forest U, 1996. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Feb 07 2009]
%H A000045 N. Renton, <a href="http://www.users.bigpond.net.au/renton/903.htm">The
fibonacci Series</a>
%H A000045 B. Rittaud, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rittaud2/
rittaud11.pdf">On the Average Growth of Random Fibonacci Sequences</
a>, Journal of Integer Sequences, 10 (2007), Article 07.2.4.
%H A000045 E. S. Rowland, <a href="http://www.math.rutgers.edu/~erowland/fibonacci.html">
Fibonacci Sequence Calculator up to n=1474</a>
%H A000045 Shiva Samieinia, <a href="http://www.math.su.se/reports/2007/6/">Digital
straight line segments and curves</a>. Licentiate Thesis. Stockholm
University, Department of Mathematics, Report 2007:6.
%H A000045 D. Schweizer, <a href="http://math.holycross.edu/~davids/fibonacci/fibonacci.html">
First 500 Fibonacci Numbers in blocks of 100.</a>
%H A000045 S. Silvia, <a href="http://arttech.about.com/library/weekly/aa060900a_fibonacci_sequence.htm">
Fibonacci sequence</a>
%H A000045 Jaap Spies, <a href="http://www.jaapspies.nl/oeis/a000045.sage">SAGE
program for computing A000045</a>
%H A000045 Z.-H. Sun, <a href="http://202.195.112.2/xsjl/szh/ConFn.pdf">Congruences
For Fibonacci Numbers</a>
%H A000045 Roberto Tauraso, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">A
New Domino Tiling Sequence</a>, Journal of Integer Sequences, Vol.
7 (2004), Article 04.2.3.
%H A000045 Thesaurus.Maths.org, <a href="http://thesaurus.maths.org/dictionary/map/
word/3788">Fibonacci sequence</a>
%H A000045 K. Tognetti, <a href="http://www.austms.org.au/Modules/Fib">Fibonacci-His
Rabbits and His Numbers and Kepler</a>
%H A000045 N. N. Vorob'ev, <a href="http://eom.springer.de/F/f040020.htm">Fibonacci
numbers</a>, Springer's Encyclopaedia of Mathematics. [From R. J.
Mathar (mathar(AT)strw.leidenuniv.nl), Nov 06 2008]
%H A000045 Carl G. Wagner, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Partition Statistics and q-Bell Numbers (q = -1)</a>, J. Integer
Seqs., Vol. 7, 2004.
%H A000045 N. P. Watson, <a href="http://www.hjnpwatson.demon.co.uk/javafibn.htm">
First 50 Fibonacci Numbers</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
FibonacciNumber.html">Link to a section of The World of Mathematics.</
a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Double-FreeSet.html">Link to a section of The World of Mathematics.</
a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Fibonaccin-StepNumber.html">Link to a section of The World of Mathematics.</
a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
ResistorNetwork.html">Link to a section of The World of Mathematics.</
a>
%H A000045 Wikipedia, <a href="http://www.wikipedia.org/wiki/Fibonacci_number">Fibonacci
number</a>
%H A000045 Willem's Fibonacci site, <a href="http://home.zonnet.nl/LeonardEuler/
fiboe.htm">Fibonacci</a>
%H A000045 G. Xiao, Numerical Calculator, <a href="http://wims.unice.fr/wims/en_tool~number~calcnum.en.html">
To display F(n), for n up to 78365,operate on "fibonacci(n)"</a>
%H A000045 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000045 <a href="Sindx_Par.html#partN">Index entries for related partition-counting
sequences</a>
%H A000045 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A000045 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A000045 G.f.: x/(1-x-x^2).
%F A000045 F(n)=((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^n*sqrt(5)).
%F A000045 Alternatively, F(n) = ((1/2+sqrt(5)/2)^n-(1/2-sqrt(5)/2)^n)/sqrt(5).
%F A000045 F(n) = F(n-1) + F(n-2) = -(-1)^n F(-n).
%F A000045 F(n) = round(phi^n/sqrt(5)).
%F A000045 F(n+1) = Sum(0 <= j <= [n/2]; binomial(n-j, j))
%F A000045 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
%F A000045 [0 1; 1 1]^n [0 1] = [F(n); F(n+1)]
%F A000045 x | F(n) ==> x | F(kn).
%F A000045 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
%F A000045 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
%F A000045 Kurmang. Aziz. Rashid (Kurmang.Rashid(AT)Btopenworld.com), Feb 21 2004,
makes 4 conjectures and gives 3 theorems:
%F A000045 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.
%F A000045 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
%F A000045 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}.
%F A000045 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).
%F A000045 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
%F A000045 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
%F A000045 |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
%F A000045 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 A000045 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
%F A000045 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 A000045 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
%F A000045 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 A000045 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
%F A000045 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 A000045 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
%F A000045 Comments from Miklos Kristof (kristmikl(AT)freemail.hu), Mar 19 2007
(Start)
%F A000045 Let L(n)=A000032=Lucas numbers. Then:
%F A000045 For a>=b and odd b, F(a+b)+F(a-b)=L(a)*F(b).
%F A000045 For a>=b and even b, F(a+b)+F(a-b)=F(a)*L(b).
%F A000045 For a>=b and odd b, F(a+b)-F(a-b)=F(a)*L(b).
%F A000045 For a>=b and even b, F(a+b)-F(a-b)=L(a)*F(b).
%F A000045 F(n+m)+(-1)^m*F(n-m)=F(n)*L(m);
%F A000045 F(n+m)-(-1)^m*F(n-m)=L(n)*F(m);
%F A000045 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 A000045 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 A000045 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 A000045 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)
%F A000045 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
%F A000045 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
%F A000045 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 A000045 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
%F A000045 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
%F A000045 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 A000045 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
%F A000045 a(n+1)=Sum_{k, 0<=k<=n} A109466(n,k)*(-1)^(n-k). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Oct 26 2008]
%F A000045 Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 25 2009:
%F A000045 a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1}
%F A000045 delta(l_1,l_2,...,l_i,...,l_n)
%F A000045 where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i + l_(i+1) >= 2 for
i=1..n-1
%F A000045 and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise.
%F A000045 sage: taylor( mul((x^2)/(1-x^2-x^4) for i in xrange(0,1)),x,0,77)#solution>
> x^2 + x^4 + 2*x^6 + 3*x^8 + 5*x^10 + 8*x^12 + 13*x^14 + 21*x^16
+ 34*x^18 + 55*x^20 + 89*x^22 + 144*x^24 + 233*x^26 + 377*x^28 +....+
514229*x^58 + 832040*x^60 + 1346269*x^62 +2178309*x^64 + 3524578*x^66
+ 5702887*x^68 + 9227465*x^70 +14930352*x^72 + 24157817*x^74 + 39088169*x^76
etc... [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01
2009]
%F A000045 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]
%e A000045 Contribution from Cino Hilliard (hillcino368(AT)gmail.com), Sep 15 2008:
(Start)
%e A000045 For x = 0,1,2,3,4 x=1/(x+1) = 1, 1/2, 2/3, 3/5, 5/8, These fractions
have
%e A000045 numerators 1,1,2,3,5 the 2nd to 6-th entries in the sequence. (End)
%p A000045 with(combinat): A000045 := proc(n) fibonacci(n); end;
%p A000045 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
%p A000045 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
%p A000045 sage: [lucas_number1(n,1,-1) for n in xrange(0, 39)] # [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%t A000045 Table[ Fibonacci[ k ], {k, 1, 50} ]
%t A000045 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]
%o A000045 (AXIOM) [fibonacci(n) for n in 0..50]
%o A000045 (MAGMA) F := func< n | Fibonacci(n) >;
%o A000045 (PARI) a(n)=fibonacci(n)
%o A000045 (PARI) a(n)=imag(quadgen(5)^n)
%o A000045 (PARI) a(n)=if(n<0,-(-1)^n*a(-n),if(n<2,n,a(n-1)+a(n-2)))
%o A000045 # Python program from Jaap Spies, Jan 05, 2007 (Change leading dots to
blanks.)
%o A000045 .def fib():
%o A000045 ... """
%o A000045 ....... generates an "infinity" of Fibonacci numbers,
%o A000045 ....... starting with 1
%o A000045 ... """
%o A000045 ... x, y = 0, 1
%o A000045 ... while 1:
%o A000045 ....... x, y = y, x+y
%o A000045 ....... yield x
%o A000045 ................
%o A000045 .f = fib()
%o A000045 .a = [f.next() for i in range(1000)] # 1000 or more
%o A000045 .a.insert(0,0)
%o A000045 ................
%o A000045 .def A000045(n):
%o A000045 ... """ returns Fibonacci number with index n, offset 0,4 """
%o A000045 ... return a[n]
%o A000045 ................
%o A000045 .def A000045_list(N):
%o A000045 ... """ returns a list of the first n Fibonacci numbers """
%o A000045 ... return a[:N]
%o A000045 ................
%o A000045 # (SAGE) Demonstration program from Jaap Spies:
%o A000045 # To see which functions are available type: sloane.A[tab]
%o A000045 # All builtin SAGE programs are called the same way:
%o A000045 # a = sloane.A000045; a # This returns the name of the sequence
%o A000045 # a(n) # This returns the n-th number of the sequence:
%o A000045 # a.list(n) # This returns a list of the first n numbers:
%o A000045 # Copy and paste the following into a worksheet or the interpreter:
%o A000045 a = sloane.A000045; print a
%o A000045 print a(0)
%o A000045 print a(1)
%o A000045 print a(2)
%o A000045 print a(38)
%o A000045 print a.list(39)
%o A000045 sage: from sage.combinat.sloane_functions import recur_gen2 sage: it
= recur_gen2(0,1,1,1) sage: [it.next() for i in range(30)] - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
%o A000045 (PARI) x=0;for(j=0,100,x=1/(x+1);print1(numerator(x)",")) [From Cino
Hilliard (hillcino368(AT)gmail.com), Sep 15 2008]
%o A000045 Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Apr 29
2009: (Start)
%o A000045 (PARI) /*Generate Fibonnaci Sequence without arrays */
%o A000045 fib(n) =
%o A000045 {
%o A000045 local(a=0,b=1);
%o A000045 print1(a","b",");
%o A000045 for(x=3,n,c=a+b;
%o A000045 print1(c",");
%o A000045 a=b;b=c;
%o A000045 );
%o A000045 }
%o A000045 (Sage) taylor( mul((x^2)/(1-x^2-x^4) for i in xrange(0,1)),x,0,77)# [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2009]
%o A000045 Contribution from Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Sep
29 2009: (Start)
%o A000045 (Haskell) Based on code
%o A000045 -- from http://www.haskell.org/haskellwiki/The_Fibonacci_sequence
%o A000045 -- which also has other versions.
%o A000045 fib :: Int -> Integer
%o A000045 fib n = fibs !! n
%o A000045 .. where
%o A000045 .... fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
%o A000045 {- Example of use: map fib [0..38] -} (End)
%Y A000045 Cf. A039834 (signed Fibonacci numbers).
%Y A000045 Cf. A000213, A000288, A000322, A000383, A060455, A030186, A039834, A020695,
A020701, A071679.
%Y A000045 Cf. A099731, A100492, A094216, A094638, A000108, A101399, A101400.
%Y A000045 First row of array A103323. Second row of array A099390.
%Y A000045 Row 2 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
%Y A000045 a(n) = A094718(4, n). a(n) = A101220(0, j, n).
%Y A000045 A000032(n)=F(n+1)+F(n-1). Cf. A060441.
%Y A000045 a(k) = A090888(0, k+1) = A118654(0, k+1) = A118654(1, k-1) = A109754(0,
k) = A109754(1, k-1), for k > 0.
%Y A000045 Cf. A059929.
%Y A000045 A144152 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]
%Y A000045 A152063 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 22 2008]
%Y A000045 Sequence in context: A132636 A152163 A039834 this_sequence A020695 A132916
A069041
%Y A000045 Adjacent sequences: A000042 A000043 A000044 this_sequence A000046 A000047
A000048
%K A000045 core,nonn,easy,nice,new
%O A000045 0,4
%A A000045 N. J. A. Sloane (njas(AT)research.att.com).
%E A000045 Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct
30 2009
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