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 A022088 Fibonacci sequence beginning 0, 5. 12
 0, 5, 5, 10, 15, 25, 40, 65, 105, 170, 275, 445, 720, 1165, 1885, 3050, 4935, 7985, 12920, 20905, 33825, 54730, 88555, 143285, 231840, 375125, 606965, 982090, 1589055, 2571145, 4160200, 6731345, 10891545, 17622890, 28514435, 46137325, 74651760, 120789085 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, pp. 15, 34, 52. LINKS Tanya Khovanova, Recursive Sequences Kristina Lund, Steven Schlicker and Patrick Sigmon, Fibonacci sequences and the space of compact sets, Involve, 1:2 (2008), pp. 159-165. Index entries for linear recurrences with constant coefficients, signature (1, 1). FORMULA a(n) = round( (2*phi-1)*phi^n ) for n>3. - Thomas Baruchel, Sep 08 2004 a(n) = 5*F(n) = L(n-1) + L(n+1) = F(n+3) + F(n-1) + F(n-4), n>3. a(n) = A119457(n+3,n-1) for n>1. - Reinhard Zumkeller, May 20 2006 G.f.: 5*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008 a(n+2) = A014217(n+4) - A014217(n). - Paul Curtz, Dec 22 2008 a(n) = sqrt(5*(A000032(n)^2-4*(-1)^n)). - Alexander Samokrutov, Sep 02 2015 From Tom Copeland, Jan 25 2016: (Start) The o.g.f. for the shifted series b(0)=0 and b(n) = a(n+1) is G(x) = 5 x(1+x)/[1-x(1+x)] = 5 L(-Cinv(-x)), where L(x) = x/(1-x) with inverse Linv(x) = x/(1+x) and Cinv(x) = x(1-x), the inverse of the o.g.f. for the shifted Catalan numbers of A000108, C(x) = (1-sqrt(1-4x))/2. Then Ginv(x) = -C(-Linv(x/5)) = [-1 + sqrt(1+4x/(5+x))]/2. a(n+1) = 5 Sum{k=0 to n} binomial(n-k,k) = 5 * A000045(n+1), from A267633, with the convention for zeros of the binomial assumed there. (End) MATHEMATICA a={}; b=0; c=5; AppendTo[a, b]; AppendTo[a, c]; Do[b=b+c; AppendTo[a, b]; c=b+c; AppendTo[a, c], {n, 1, 12, 1}]; a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *) LinearRecurrence[{1, 1}, {0, 5}, 40] (* Harvey P. Dale, Jan 13 2012 *) PROG (MAGMA) [5*Fibonacci(n): n in [1..40]]; // Vincenzo Librandi, Sep 03 2015 (PARI) a(n) = 5*fibonacci(n); \\ Michel Marcus, Sep 03 2015 CROSSREFS Cf. A000032, A000045. Cf. A000108, A267633. Sequence in context: A000728 A242895 A242129 * A245418 A082450 A304266 Adjacent sequences:  A022085 A022086 A022087 * A022089 A022090 A022091 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 18 16:28 EDT 2018. Contains 316323 sequences. (Running on oeis4.)