%I #14 Sep 08 2022 08:45:32
%S 1,17,260,516,3641,4937,21744,25840,84889,94889,255940,276676,647969,
%T 686385,1445760,1511296,2931153,3036129,5512228,5672228,9756329,
%U 9990585,16426928,16758704,26524329,26981305,41330212,41944868,62456017
%N a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n^4 if n is even.
%H G. C. Greubel, <a href="/A135214/b135214.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).
%F From _R. J. Mathar_, May 17 2008: (Start)
%F O.g.f.: x*(1 + x^2)*(x^8 - 16*x^7 + 236*x^6 - 144*x^5 + 1446*x^4 + 144*x^3 + 236*x^2 + 16*x + 1)/((1-x)^7 *(1+x)^6).
%F a(2*n-1) = n*(-8 + 80*n^2 + 48*n^4 + 80*n^5 + 35*n - 220*n^3)/15.
%F a(2*n) = n*(-8 + 80*n^2 + 48*n^4 + 80*n^5 + 35*n + 20*n^3)/15 . (End)
%t a = {}; r = 5; s = 4; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a
%t LinearRecurrence[{1, 6, -6, -15, 15, 20, -20, -15, 15, 6, -6, -1, 1}, {1, 17, 260, 516, 3641, 4937, 21744, 25840, 84889, 94889, 255940, 276676, 647969}, 50] (* _G. C. Greubel_, Oct 04 2016 *)
%o (PARI) x='x+O('x^50); Vec(x*(1 + x^2)*(x^8 - 16*x^7 + 236*x^6 - 144*x^5 + 1446*x^4 + 144*x^3 + 236*x^2 + 16*x + 1)/((1-x)^7 *(1+x)^6)) \\ _G. C. Greubel_, Jul 04 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1 + x^2)*(x^8 - 16*x^7 + 236*x^6 - 144*x^5 + 1446*x^4 + 144*x^3 + 236*x^2 + 16*x + 1)/((1-x)^7 *(1+x)^6))) // _G. C. Greubel_, Jul 04 2018
%Y Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.
%K nonn
%O 1,2
%A _Artur Jasinski_, May 12 2008