Search: id:A000045 Results 1-1 of 1 results found. %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 00, 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, Missouri Journal of Mathematical Sciences, Vol. 2, No. 2, Spring 1990, pp. 78-79. Zentralblatt MATH, Zbl 1097.11516. %D A000045 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. %D A000045 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 70. %D A000045 R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventues in Applied Mathematics, Princeton Univ. Press, 1999. 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Merrill, The Fib-Phi Link Page %H A000045 Jean-Christophe Michel, Le nombre d'or dans l'ensemble de Mandelbrot (in French, 'The golden number in the Mandelbrot set') %H A000045 H. Mishima, Factorizations of many number sequences %H A000045 H. Mishima, Factorizations of many number sequences %H A000045 H. Mishima, Factorizations of many number sequences %H A000045 H. Mishima, Factorizations of many number sequences %H A000045 H. Mishima, Factorizations of many number sequences %H A000045 P. Moree, Convoluted convolved Fibonacci numbers %H A000045 Newton's Institute, Posters in the London Underground %H A000045 J. Patterson, The Fibonacci Sequence %H A000045 Ivars Peterson, Fibonacci's Missing Flowers. %H A000045 S. Plouffe, Project Gutenberg, The First 1001 Fibonacci Numbers %H A000045 S. Plouffe, Fibonacci numbers [Contains the first 754965 terms] %H A000045 S. Rabinowitz, Algorithmic Manipulation of Fibonacci Identities (1996). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 06 2008] %H A000045 Marc Renault, Properties of the Fibonacci sequence under various moduli, MSc Thesis, Wake Forest U, 1996. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 07 2009] %H A000045 N. Renton, The fibonacci Series %H A000045 B. Rittaud, On the Average Growth of Random Fibonacci Sequences, Journal of Integer Sequences, 10 (2007), Article 07.2.4. %H A000045 E. S. Rowland, Fibonacci Sequence Calculator up to n=1474 %H A000045 Shiva Samieinia, Digital straight line segments and curves. Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6. %H A000045 D. Schweizer, First 500 Fibonacci Numbers in blocks of 100. %H A000045 S. Silvia, Fibonacci sequence %H A000045 Jaap Spies, SAGE program for computing A000045 %H A000045 Z.-H. Sun, Congruences For Fibonacci Numbers %H A000045 Roberto Tauraso, A New Domino Tiling Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.3. %H A000045 Thesaurus.Maths.org, Fibonacci sequence %H A000045 K. Tognetti, Fibonacci-His Rabbits and His Numbers and Kepler %H A000045 N. N. Vorob'ev, Fibonacci numbers, Springer's Encyclopaedia of Mathematics. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 06 2008] %H A000045 Carl G. Wagner, Partition Statistics and q-Bell Numbers (q = -1), J. Integer Seqs., Vol. 7, 2004. %H A000045 N. P. Watson, First 50 Fibonacci Numbers %H A000045 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000045 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000045 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000045 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000045 Wikipedia, Fibonacci number %H A000045 Willem's Fibonacci site, Fibonacci %H A000045 G. Xiao, Numerical Calculator, To display F(n), for n up to 78365,operate on "fibonacci(n)" %H A000045 Index entries for "core" sequences %H A000045 Index entries for related partition-counting sequences %H A000045 Index entries for two-way infinite sequences %H A000045 Index entries for sequences related to linear recurrences with constant coefficients %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= 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 Search completed in 0.017 seconds