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A001628 Convolved Fibonacci numbers.
(Formerly M2789 N1124)
24
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)*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. - 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

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).

LINKS

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.

V. E. Hoggatt, Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.

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

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 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.

Index entries for sequences related to linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

FORMULA

G.f.: 1 / (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

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 *)

PROG

(PARI) Vec((1 - x - x^2 )^-3+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011

CROSSREFS

a(n) = A037027(n+2,2) (Fibonacci convolution triangle).

Cf. A055243.

Sequence in context: A121589 A227454 A000716 * A099166 A222083 A202882

Adjacent sequences:  A001625 A001626 A001627 * A001629 A001630 A001631

KEYWORD

easy,nonn

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified September 1 07:55 EDT 2014. Contains 246289 sequences.