A024313 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3), t = A023531.

3, 3, 10, 17, 37, 59, 114, 185, 334, 540, 938, 1518, 2573, 4163, 6946, 11239, 18559, 30029, 49248, 79685, 130090, 210490, 342596, 554332, 900423, 1456915, 2363370, 3824013, 6197753

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A024312, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000032, A000045, A023531.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( x*(3-2*x^2+4*x^3-x^4-3*x^5-2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) )); // G. C. Greubel, Jan 17 2022

Mathematica

a[n_]:= With[{F=Fibonacci}, If[EvenQ[n], LucasL[n+3] + F[n+2] - LucasL[n/2 +3] - (n/2 +1)*F[n/2 +2], LucasL[n+3] + F[n+2] - LucasL[(n+5)/2]-(n+3)/2*Fibonacci[(n+3)/2]]];
Table[a[n], {n, 40}] (* G. C. Greubel, Jan 17 2022 *)

Sage

def A024313_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(3 -2*x^2 +4*x^3 -x^4 -3*x^5 -2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2) ).list()
a=A024313_list(41); a[1:] # G. C. Greubel, Jan 17 2022

Formula

G.f.: x*(3-2*x^2+4*x^3-x^4-3*x^5-2*x^7)/((1-x-x^2)*(1-x^2-x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

From G. C. Greubel, Jan 17 2022: (Start)

a(2*n) = Lucas(2*n+3) + F(2*n+2) - Lucas(n+3) - (n+1)*F(n+2).

a(2*n+1) = Lucas(2*n+4) + F(2*n+3) - Lucas(n+3) - (n+2)*F(n+2), where F(n) = A000045(n). (End)