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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).
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%I #15 Oct 21 2022 22:12:01

%S 3,6,17,27,59,96,185,299,540,874,1518,2456,4163,6736,11239,18185,

%T 30029,48588,79685,128933,210490,340580,554332,896928,1456915,2357338,

%U 3824013,6187383

%N 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).

%H G. C. Greubel, <a href="/A024315/b024315.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2,-1,-1,-3,2,1,1,1).

%F 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

%F From _G. C. Greubel_, Jan 16 2022: (Start)

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

%F 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)

%t 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]]];

%t Table[a[n], {n, 40}] (* _G. C. Greubel_, Jan 16 2022 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 40);

%o 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

%o (Sage)

%o def A024315_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o 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()

%o a=A024315_list(41); a[1:] # _G. C. Greubel_, Jan 16 2022

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

%Y Cf. A000032, A000045.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_