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%I M2733 #98 Jul 24 2022 02:53:25
%S 0,0,0,1,3,8,18,38,76,147,277,512,932,1676,2984,5269,9239,16104,27926,
%T 48210,82900,142055,242665,413376,702408,1190808,2014608,3401833,
%U 5734251,9650312,16216602,27213182,45608092,76345851,127656829,213230144,355817324,593205284
%N a(n) = a(n-1) + a(n-2) + F(n) - 1, a(0) = a(1) = 0, where F() = Fibonacci numbers A000045.
%C Partial sums of A001629.
%C Number of edges in the Fibonacci hypercube FQ(n-2) (defined in the Rispoli and Cosares reference). - _Emeric Deutsch_, Oct 06 2014
%C Circuit rank (cyclomatic number) of the n-Fibonacci cube graph. - _Eric W. Weisstein_, Sep 05 2017
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006478/b006478.txt">Table of n, a(n) for n = 0..1000</a>
%H Carlos Alirio Rico Acevedo and Ana Paula Chaves, <a href="https://arxiv.org/abs/1903.07490">Double-Recurrence Fibonacci Numbers and Generalizations</a>, arXiv:1903.07490 [math.NT], 2019.
%H Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/2010.09505">Gray codes for Fibonacci q-decreasing words</a>, arXiv:2010.09505 [cs.DM], 2020.
%H Sergey Kirgizov, <a href="https://arxiv.org/abs/2201.00782">Q-bonacci words and numbers</a>, arXiv:2201.00782 [math.CO], 2022.
%H K. J. Overholt, <a href="http://dx.doi.org/10.1007/BF01933527">Efficiency of the Fibonacci search method</a>, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 92-96.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H F. J. Rispoli and S. Cosares, <a href="http://ajc.maths.uq.edu.au/pdf/40/ajc_v40_p187.pdf">The Fibonacci hypercube</a>, Australasian J. Combinatorics, 40, 2008, 187-196.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CircuitRank.html">Circuit Rank</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciCubeGraph.html">Fibonacci Cube Graph</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,1,1).
%F a(n) - a(n-1) = A001629(n-1).
%F a(n) = 1 + ((n-5)*F(n-1) + (3*n-8)*F(n))/5.
%F G.f.: x^3/((1-x)*(1-x-x^2)^2). - _Simon Plouffe_ in his 1992 dissertation
%F a(n) = Sum_{k=0..n-1} Sum_{i=0..k} F(i)*F(k-i). - _Benoit Cloitre_, Jan 26 2003
%F a(n) = A175722(-2-n). - _Michael Somos_, Mar 11 2014
%F 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
%F 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
%e 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 + ...
%p A006478 := proc(n)
%p 1 + ((n-5)*combinat[fibonacci](n-1)+(3*n-8)*combinat[fibonacci](n)) / 5;
%p end proc:
%p seq(A006478(n),n=0..20) ; # _R. J. Mathar_, Jun 12 2018
%t CoefficientList[Series[x^3/((1 - x) (1 - x - x^2)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 13 2014 *)
%t LinearRecurrence[{3, -1, -3, 1, 1}, {0, 0, 0, 1, 3, 8}, 20] (* _Eric W. Weisstein_, Sep 05 2017 *)
%t Table[1 + (2 (n + 1) Fibonacci[n] + n Fibonacci[n + 1])/5 - Fibonacci[n + 2], {n, 0, 20}] (* _Eric W. Weisstein_, Sep 05 2017 *)
%o (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 */
%o (Haskell)
%o a006478 n = a006478_list !! (n-3)
%o a006478_list = scanl1 (+) $ drop 2 a001629_list
%o -- _Reinhard Zumkeller_, Sep 12 2015
%Y Cf. A000045, A000051, A006479, A175722.
%Y Cf. A001629.
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_
%E a(0)-a(2) added and offset changed - _N. J. A. Sloane_, Jun 19 2021
%E Programs and b-file adapted by _Georg Fischer_, Jun 21 2021
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