OFFSET
0,2
REFERENCES
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
T. D. Noe, Table of n, a(n) for n = 0..500
Daniel Birmajer, Juan Gil and Michael D. Weiner, Linear recurrence sequences and their convolutions via Bell polynomials, arXiv:1405.7727 [math.CO], 2014.
Daniel Birmajer, Juan Gil and Michael D. Weiner, Linear Recurrence Sequences and Their Convolutions via Bell Polynomials, Journal of Integer Sequences, 18 (2015), #15.1.2.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Verner E. Hoggatt, Jr. and Marjorie Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.
Milan Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8, section 3.
Toufik Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.
Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 18.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,5,8,-2,-4,-1).
FORMULA
G.f.: 1/(1 - x - x^2)^4.
a(n) = A037027(n+3, 3) (Fibonacci convolution triangle).
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, 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, 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, Jan 18 2006
a(n) = (((-i)^n)/3!)*(d^3/dx^3)S(n+3,x)|_{x=i}, where i is the imaginary unit. Third derivative of Chebyshev S(n+3,x) polynomial evaluated at x=i multiplied by ((-i)^(n-3))/3!. See A049310 for the S-polynomials. - Wolfdieter Lang, Apr 04 2007
a(n) = Sum_{i=ceiling(n/2)..n} (i+1)*(i+2)*(i+3)*binomial(i,n-i)/6. - Vladimir Kruchinin, Apr 26 2011
Recurrence: a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 5*a(n-4) + 8*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8). - Fung Lam, May 11 2014
n*a(n) - (n+3)*a(n-1) - (n+6)*a(n-2) = 0, n > 1. - Michael D. Weiner, Nov 18 2014
MAPLE
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); # Alois P. Heinz, Aug 15 2008
MATHEMATICA
CoefficientList[Series[1/(1 - x - x^2)^4, {x, 0, 100}], x] (* Stefan Steinerberger, Apr 15 2006 *)
PROG
(PARI) Vec( 1/(1 - x - x^2)^4 + O(x^66) ) \\ Joerg Arndt, May 12 2014
(Magma) [(n+5)*(n+3)*(4*(n+1)*Fibonacci(n+2)+3*(n+2)*Fibonacci(n+1))/150: n in [0..30]]; // Vincenzo Librandi, Nov 19 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved