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%I #14 Sep 08 2022 08:45:39
%S 0,0,1,63,665,3368,11592,31696,74361,156087,300993,542920,927648,
%T 1515416,2383745,3630375,5376505,7770336,10990728,15251160,20803993,
%U 27944847,37017281,48417776,62600832,80084368,101455425,127375983
%N a(n) + a(n+1) + a(n+2) = n^6, a(1)=a(2)=0.
%C 0 + 0 + 1 = 1^6; 0 + 1 + 63 = 2^6; ...
%H G. C. Greubel, <a href="/A152731/b152731.txt">Table of n, a(n) for n = 1..5000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,21,-21,21,-21,15,-6,1).
%F From _R. J. Mathar_, Dec 12 2008: (Start)
%F a(n) = -26*n/3 + 20*n^3/3 - 5*n^2 + 7/3 - 2*n^5 + n^6/3 + 5*n^4/3 - 7*A131713(n)/3.
%F G.f.: x^3*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1)/((1-x)^7*(1+x+x^2)). (End)
%t k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^6-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
%t LinearRecurrence[{6, -15, 21, -21, 21, -21, 15, -6, 1}, {0, 0, 1, 63, 665, 3368, 11592, 31696, 74361}, 5000]
%t CoefficientList[Series[x^2*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1)/((1-x)^7*(1+x+x^2)),{x, 0, 5000}], x] (* _Stefano Spezia_, Sep 02 2018 *)
%o (PARI) x='x+O('x^30); concat([0,0], Vec(x^3*(1+x)*(x^4+56*x^3 +246*x^2 +56*x+1)/((1-x)^7*(1+x+x^2)))) \\ _G. C. Greubel_, Sep 01 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^3*(1+x)*(x^4+56*x^3+246*x^2+56*x+1)/((1-x)^7*(1 +x+ x^2)))); // _G. C. Greubel_, Sep 01 2018
%Y Cf. A152728, A152729, A152730, A152725, A152726, A000212.
%K nonn
%O 1,4
%A _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008
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