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 A001628 Convolved Fibonacci numbers. (Formerly M2789 N1124) 33
 1, 3, 9, 22, 51, 111, 233, 474, 942, 1836, 3522, 6666, 12473, 23109, 42447, 77378, 140109, 252177, 451441, 804228, 1426380, 2519640, 4434420, 7777860, 13599505, 23709783, 41225349, 71501422, 123723351, 213619683, 368080793, 633011454, 1086665562, 1862264196 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n-2) = (((-i)^(n-2))/2)*(d^2/dx^2)S(n,x)|_{x=i}, n>=2. Second derivative of Chebyshev S-polynomials evaluated at x=i (imaginary unit) multiplied by ((-i)^(n-2))/2. See A049310 for the S-polynomials. - Wolfdieter Lang, Apr 04 2007 a(n) = number of weak compositions of n in which exactly 2 parts are 0 and all other parts are either 1 or 2. - Milan Janjic, Jun 28 2010 Number of 4-cycles in the Fibonacci cube Gamma[n+3] (see the Klavzar reference, p. 511. - Emeric Deutsch, Apr 17 2014 REFERENCES T. Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001, p. 375. J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS G. C. Greubel, Table of n, a(n) for n = 0..4500 (terms 0..500 from T. D. Noe) D. Birmajer, J. Gil and M. Weiner, Linear recurrence sequences and their convolutions via Bell polynomials, arXiv:1405.7727 [math.CO], 2014. D. Birmajer, J. B. Gil, M. D. Weiner, Linear Recurrence Sequences and Their Convolutions via Bell Polynomials, Journal of Integer Sequences, 18 (2015), #15.1.2. P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. V. E. Hoggatt, Jr., Letters to N. J. A. Sloane, 1974-1975 V. E. Hoggatt, Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122. M. Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8, section 3. S. Klavzar, Structure of Fibonacci cubes: a survey, preprint. S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522 T. Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003. P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003. Pieter Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Jeffrey B. Remmel, J. L. B. Tiefenbruck, Q-analogues of convolutions of Fibonacci numbers, Australasian Journal of Combinatorics, Volume 64(1) (2016), Pages 166-193. J. Riordan, Letter to N. J. A. Sloane, Oct. 1970 Eric Weisstein's World of Mathematics, Graph Cycle Eric Weisstein's World of Mathematics, Fibonacci Cube Graph Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1). FORMULA G.f.: 1 / (1 - x - x^2)^3. a(n) = A037027(n+2,2) (Fibonacci convolution triangle). a(n) = ((5*n+16)*(n+1)*F(n+2)+(5*n+17)*(n+2)*F(n+1))/50, F=A000045. -Wolfdieter Lang, Apr 12 2000 (This formula coincides with eq. (32.14) of the Koshy reference, p. 375, if there n -> n+3. - Wolfdieter Lang, Aug 03 2012) For n>2, a(n-2) = sum(i+j+k=n, F(i)*F(j)*F(k)). - Benoit Cloitre, Nov 01 2002 a(n) = F''(n+2, 1)/2, i.e. 1/2 times the 2nd derivative of the (n+2)th Fibonacci polynomial evaluated at 1. - T. D. Noe, Jan 18 2006 a(n) = Sum_{k=0..n} C(k,n-k)*C(k+2,2). - Paul Barry, Apr 13 2008 0 = n*a(n) - (n+2)*a(n-1) - (n+4)*a(n-2), n>1. - Michael D. Weiner, Nov 18 2014 a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6). - Eric W. Weisstein, Sep 05 2017 EXAMPLE G.f. = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 111*x^5 + 233*x^6 + 474*x^7 + ... MAPLE A001628:=-1/(z**2+z-1)**3; [Simon Plouffe in his 1992 dissertation.] a:= n-> (Matrix(6, (i, j)-> `if` (i=j-1, 1, `if` (j=1, [3, 0, -5, 0, 3, 1][i], 0)))^n)[1, 1]: seq (a(n), n=0..29); # Alois P. Heinz, Aug 01 2008 MATHEMATICA CoefficientList[Series[1/(-z^2 - z + 1)^3, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *) (* Start Eric W. Weisstein, Sep 05 2017 *) Table[Derivative[2][Fibonacci[n + 2, #] &][1]/2, {n, 20}] Derivative[2][Fibonacci[Range[20] + 2, #] &][1]/2 LinearRecurrence[{3, 0, -5, 0, 3, 1}, {1, 3, 9, 22, 51, 111}, 20] Table[-I^(n + 1) Derivative[2][ChebyshevU[n + 1, -#/2] &][I]/2, {n, 20}] (* End *) PROG (PARI) Vec((1 - x - x^2 )^-3+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011 (MAGMA) [(&+[Binomial(k, n-k)*Binomial(k+2, 2): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 10 2018 CROSSREFS a(n) = A037027(n+2,2) (Fibonacci convolution triangle). Cf. A055243. Cf. A291915 (6-cycles). Sequence in context: A121589 A227454 A000716 * A099166 A222083 A305612 Adjacent sequences:  A001625 A001626 A001627 * A001629 A001630 A001631 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified December 12 19:33 EST 2018. Contains 318081 sequences. (Running on oeis4.)