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%I #23 Jan 06 2024 00:59:37
%S 0,0,1,15,65,176,384,736,1281,2079,3201,4720,6720,9296,12545,16575,
%T 21505,27456,34560,42960,52801,64239,77441,92576,109824,129376,151425,
%U 176175,203841,234640,268800,306560,348161,393855,443905,498576,558144
%N a(n) = (n-2)^4 - a(n-1) - a(n-2), with a(1) = a(2) = 0.
%C a(n+2) - a(n-1) = n^4 - (n-1)^4 = A005917(n) for all n in Z. - _Michael Somos_, Sep 02 2018
%H G. C. Greubel, <a href="/A152729/b152729.txt">Table of n, a(n) for n = 1..5000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,5,-5,6,-4,1).
%F G.f.: -x^3*(x+1)*(x^2+10*x+1) / ((x-1)^5*(x^2+x+1)). - _Colin Barker_, Oct 28 2014
%F a(n) = a(2 - n) for all n in Z. - _Michael Somos_, Sep 02 2018
%e 0 + 0 + 1 = 1^4; 0 + 1 + 15 = 2^4; 1 + 15 + 65 = 3^4; ...
%e G.f. = x^3 + 15*x^4 + 65*x^5 + 176*x^6 + 384*x^7 + 736*x^8 + 1281*x^9 + ... - _Michael Somos_, Sep 02 2018
%t k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^4-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,4!}];lst
%t LinearRecurrence[{4,-6,5,-5,6,-4,1}, {0,0,1,15,65,176,384}, 50] (* _G. C. Greubel_, Sep 01 2018 *)
%t a[ n_] := With[ {m = Max[n, 2 - n]}, SeriesCoefficient[ x^3 (1 + x) (1 + 10 x + x^2) / ((1 - x)^5 (1 + x + x^2)), {x , 0, m}]]; (* _Michael Somos_, Sep 02 2018 *)
%o (PARI) concat([0,0], Vec(-x^3*(x+1)*(x^2+10*x+1)/((x-1)^5*(x^2+x+1)) + O(x^100))) \\ _Colin Barker_, Oct 28 2014
%o (PARI) {a(n) = my(m = max(n, 2 - n)); polcoeff( x^3 * (1 + x) * (1 + 10*x + x^2) / ((1 - x)^5 * (1 + x + x^2)) + x * O(x^m), m)}; /* _Michael Somos_, Sep 02 2018 */
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^3*(x+1)*(x^2+10*x+1)/((1-x)^5*(x^2+x+1)))); // _G. C. Greubel_, Sep 01 2018
%Y Cf. A005917, A152728, A152725, A152726, A000212.
%K nonn,easy
%O 1,4
%A _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008
%E Definition adapted to offset by _Georg Fischer_, Jun 18 2021
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