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A001628
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Convolved Fibonacci numbers.
(Formerly M2789 N1124)
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35
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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G.f. = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 111*x^5 + 233*x^6 + 474*x^7 + ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Magma) [(&+[Binomial(k, n-k)*Binomial(k+2, 2): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 10 2018
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CROSSREFS
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a(n) = A037027(n+2,2) (Fibonacci convolution triangle).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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