A024315 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = floor(n/2), s = (natural numbers >= 3), t = (Fibonacci numbers).

3, 6, 17, 27, 59, 96, 185, 299, 540, 874, 1518, 2456, 4163, 6736, 11239, 18185, 30029, 48588, 79685, 128933, 210490, 340580, 554332, 896928, 1456915, 2357338, 3824013, 6187383

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1,-1,-3,2,1,1,1).

Formula

G.f.: x*(3 +3*x +2*x^2 -2*x^3 -4*x^4 -x^5 -2*x^6)/((1-x-x^2)*(1-x^2-x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

From G. C. Greubel, Jan 16 2022: (Start)

a(2*n) = L(2*n+4) + F(2*n+3) - F(n+5) - (n+2)*F(n+3), n >= 1.

a(2*n-1) = L(2*n+3) + F(2*n+2) - F(n+3) - (n+3)*F(n+2), n >= 1, where L(n) = A000032(n) and F(n) = A000045(n). (End)

Mathematica

a[n_]:= With[{F=Fibonacci}, If[EvenQ[n], LucasL[n+4] +F[n+3] -F[(n+10)/2] -((n+ 4)/2)*F[(n+6)/2], LucasL[n+4] +F[n+3] -F[(n+7)/2] -((n+7)/2)*F[(n+5)/2]]];
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 16 2022 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( x*(3+3*x+2*x^2-2*x^3-4*x^4-x^5-2*x^6)/((1-x-x^2)*(1-x^2-x^4)^2) )); // G. C. Greubel, Jan 16 2022
(Sage)
def A024315_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x*(3+3*x+2*x^2-2*x^3-4*x^4-x^5-2*x^6)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()
a=A024315_list(41); a[1:] # G. C. Greubel, Jan 16 2022

Crossrefs

Cf. A024312, A024313, A024314, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000032, A000045.

Sequence in context: A063618 A217084 A024823 * A235088 A369707 A327068

Adjacent sequences: A024312 A024313 A024314 * A024316 A024317 A024318

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved