0,2

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

P. Moree, Convoluted convolved Fibonacci numbers

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

G.f.: 1/(1 - x - x^2)^4.

a(n)= (n+5)*(n+3)*(4*(n+1)*F(n+2)+3*(n+2)*F(n+1))/150, F(n)=A000045(n). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 12 2000

For n>3, a(n-3) = sum(h+i+j+k=n, F(h)*F(i)*F(j)*F(k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 01 2002

a(n)=F'''(n+3, 1)/6, i.e. 1/6 times the 3rd derivative of the (n+3)th Fibonacci polynomial evaluated at 1. - T. D. Noe (noe(AT)sspectra.com), Jan 18 2006

a(n)=(((-I)^n)/3!)*diff(S(n+3,x),x$3)|_{x=I}. Third derivative of Chebyshev S(n+3,x) polynomial evaluated at x=I (imaginary unit) multiplied by ((-I)^(n-3))/3!. See A049310 for the S-polynomials. W. Lang, Apr 04 2007

sage: taylor( mul(x^2/(1-x-x^2)^2 for i in xrange(0,2)),x,0,33)# solution>> x^4 + 4*x^5 + 14*x^6 + 40*x^7 + 105*x^8 + 256*x^9 + 594*x^10 + 1324*x^11 + 2860*x^12 + 6020*x^13 + 12402*x^14 +......+ 93914325*x^28 + 169457708*x^29 + 304597382*x^30 + 545556512*x^31 + 973877245*x^32 + 1733053440*x^33, etc... [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2009]

a := n-> (Matrix(8, (i, j)-> if (i=j-1) then 1 elif j=1 then [4, -2, -8, 5, 8, -2, -4, -1][i] else 0 fi)^n)[1, 1]; seq (a(n), n=0..29); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 15 2008]

CoefficientList[Series[1/(1 - x - x^2)^4, {x, 0, 100}], x] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 15 2006

(Other) sage: taylor( mul(x^2/(1-x-x^2)^2 for i in xrange(0, 2)), x, 0, 33)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2009]

a(n)= A037027(n+3, 3) (Fibonacci convolution triangle).

Sequence in context: A066368 A121593 A023003 this_sequence A054443 A072674 A032285

Adjacent sequences: A001869 A001870 A001871 this_sequence A001873 A001874 A001875

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 08 2000