A089098 Sign twisted convoluted convolved Fibonacci numbers H_j^(2).

1, 1, 3, 5, 11, 19, 37, 65, 120, 210, 376, 654, 1149, 1985, 3443, 5911, 10159, 17345, 29605, 50305, 85400, 144516, 244272, 411900, 693729, 1166209, 1958219, 3283145, 5498595, 9197455, 15369373, 25655489, 42787456, 71293590, 118695272, 197452746, 328227725

Offset: 1

Author

N. J. A. Sloane, Dec 05 2003

Keywords

• nonn
• easy

Comments

Let "a" = the Fibonacci numbers, and "b" = the aerated Fibonacci numbers (1, 0, 1, 0, 2,...). Then A089098 = (1/2) * (a^2 + b), where a^2 = A001629, the Fibonacci numbers convolved with themselves: (1, 2, 5, 10, 20, 38,...).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, Vertex and edge orbits of Fibonacci and Lucas cubes, 2014; See Table 2.
• P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-4,-1,0,0,2,1).

Crossrefs

2nd column of A337009.

Programs

Maple

with(numtheory): f := z->-1/(1-z-z^2): m := proc(r,j) d := divisors(r): W := (1/r)*z*sum(mobius(d[i])*f(z^d[i])^(r/d[i]),i=1..nops(d)): Wser := simplify(series(W,z=0,80)): coeff(Wser,z^j) end: seq(m(2,j),j=1..39);

Mathematica

(1-x-x^2+x^3)/((1-x-x^2)^2*(1-x^2-x^4)) + O[x]^40 // CoefficientList[#,x]& (* Jean-François Alcover, Jan 20 2018 *)

PARI

Vec(-x*(x-1)^2*(x+1)/((x^2+x-1)^2*(x^4+x^2-1)) + O(x^50)) \\ Colin Barker, Jul 23 2015

Formula

G.f.: (z/2)[1/(1-z-z^2)^2+1/(1-z^2-z^4)].

G.f.: -x*(x-1)^2*(x+1) / ((x^2+x-1)^2*(x^4+x^2-1)). - Colin Barker, Jul 23 2015

Extensions

Edited by Emeric Deutsch, Mar 06 2004