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A001628
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Convolved Fibonacci numbers.
(Formerly M2789 N1124)
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21
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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)*diff(S(n,x),x$2)|_{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. [From Milan Janjic, Jun 28 2010]
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REFERENCES
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V. E. Hoggatt, Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.
T. Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001, p. 375
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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T. D. Noe, Table of n, a(n) for n=0..500
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
T. Mansour, Generalization of some identities involving the Fibonacci numbers
P. Moree, Convoluted convolved Fibonacci numbers
Pieter Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).
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FORMULA
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G.f.: ( 1 - x - x^2 )^-3.
a(n) = ((5*n+16)*(n+1)*F(n+2)+(5*n+17)*(n+2)*F(n+1))/50, F(n)=A000045(n). -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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MAPLE
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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
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MATHEMATICA
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CoefficientList[Series[1/(-z^2 - z + 1)^3, {z, 0, 100}], z] (* From Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
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PROG
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(PARI) a(n)=Vec( 1 - x - x^2 )^-3+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011
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CROSSREFS
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a(n)= A037027(n+2, 2) (Fibonacci convolution triangle).
Cf. A055243.
Sequence in context: A160526 A121589 A000716 * A099166 A222083 A202882
Adjacent sequences: A001625 A001626 A001627 * A001629 A001630 A001631
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KEYWORD
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easy,nonn,changed
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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STATUS
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approved
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