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 A006478 a(n) = a(n-1) + a(n-2) + F(n) - 1, a(0) = a(1) = 0, where F() = Fibonacci numbers A000045. (Formerly M2733) 13
 0, 0, 0, 1, 3, 8, 18, 38, 76, 147, 277, 512, 932, 1676, 2984, 5269, 9239, 16104, 27926, 48210, 82900, 142055, 242665, 413376, 702408, 1190808, 2014608, 3401833, 5734251, 9650312, 16216602, 27213182, 45608092, 76345851, 127656829, 213230144, 355817324, 593205284 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Partial sums of A001629. Number of edges in the Fibonacci hypercube FQ(n-2) (defined in the Rispoli and Cosares reference). - Emeric Deutsch, Oct 06 2014 Circuit rank (cyclomatic number) of the n-Fibonacci cube graph. - Eric W. Weisstein, Sep 05 2017 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Carlos Alirio Rico Acevedo and Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019. Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Gray codes for Fibonacci q-decreasing words, arXiv:2010.09505 [cs.DM], 2020. Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022. K. J. Overholt, Efficiency of the Fibonacci search method, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 92-96. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 F. J. Rispoli and S. Cosares, The Fibonacci hypercube, Australasian J. Combinatorics, 40, 2008, 187-196. Eric Weisstein's World of Mathematics, Circuit Rank Eric Weisstein's World of Mathematics, Fibonacci Cube Graph Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1). FORMULA a(n) - a(n-1) = A001629(n-1). a(n) = 1 + ((n-5)*F(n-1) + (3*n-8)*F(n))/5. G.f.: x^3/((1-x)*(1-x-x^2)^2). - Simon Plouffe in his 1992 dissertation a(n) = Sum_{k=0..n-1} Sum_{i=0..k} F(i)*F(k-i). - Benoit Cloitre, Jan 26 2003 a(n) = A175722(-2-n). - Michael Somos, Mar 11 2014 a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5). - Eric W. Weisstein, Sep 05 2017 E.g.f.: exp(x) + exp(x/2)*(5*(3*x - 5)*cosh(sqrt(5)*x/2) + sqrt(5)*(5*x - 11)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Jul 24 2022 EXAMPLE G.f. = x^3 + 3*x^4 + 8*x^5 + 18*x^6 + 38*x^7 + 76*x^8 + 147*x^9 + 277*x^10 + ... MAPLE A006478 := proc(n) 1 + ((n-5)*combinat[fibonacci](n-1)+(3*n-8)*combinat[fibonacci](n)) / 5; end proc: seq(A006478(n), n=0..20) ; # R. J. Mathar, Jun 12 2018 MATHEMATICA CoefficientList[Series[x^3/((1 - x) (1 - x - x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 13 2014 *) LinearRecurrence[{3, -1, -3, 1, 1}, {0, 0, 0, 1, 3, 8}, 20] (* Eric W. Weisstein, Sep 05 2017 *) Table[1 + (2 (n + 1) Fibonacci[n] + n Fibonacci[n + 1])/5 - Fibonacci[n + 2], {n, 0, 20}] (* Eric W. Weisstein, Sep 05 2017 *) PROG (PARI) {a(n) = if( n<0, polcoeff( x^2 / ((1 - x) * (1 + x - x^2)^2) + x * O(x^-n), -n), polcoeff( x^3 / ((1 - x) * (1 - x - x^2)^2) + x * O(x^n), n))}; /* Michael Somos, Mar 11 2014 */ (Haskell) a006478 n = a006478_list !! (n-3) a006478_list = scanl1 (+) \$ drop 2 a001629_list -- Reinhard Zumkeller, Sep 12 2015 CROSSREFS Cf. A000045, A000051, A006479, A175722. Cf. A001629. Sequence in context: A078409 A036642 A000235 * A104187 A131051 A051633 Adjacent sequences: A006475 A006476 A006477 * A006479 A006480 A006481 KEYWORD nonn,easy AUTHOR N. J. A. Sloane EXTENSIONS a(0)-a(2) added and offset changed - N. J. A. Sloane, Jun 19 2021 Programs and b-file adapted by Georg Fischer, Jun 21 2021 STATUS approved

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Last modified December 4 11:57 EST 2023. Contains 367560 sequences. (Running on oeis4.)