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%I #25 Dec 29 2022 09:08:25
%S 0,0,1,3,6,10,15,21,28,36,46,58,72,88,106,126,148,172,199,229,262,298,
%T 337,379,424,472,524,580,640,704,772,844,920,1000,1085,1175,1270,1370,
%U 1475,1585,1700,1820,1946,2078,2216,2360,2510,2666,2828,2996,3171,3353,3542,3738
%N Expansion of g.f. x^2/((1 - x)^4*(1 + x)*(1 + x^2)*(1 + x^4)).
%H G. C. Greubel, <a href="/A122056/b122056.txt">Table of n, a(n) for n = 0..1000</a>
%H A. N. W. Hone, <a href="http://arXiv.org/abs/0807.2538">Algebraic curves, integer sequences and a discrete Painlevé transcendent</a>, arXiv:0807.2538 [nlin.SI], 2008; Proceedings of SIDE 6, Helsinki, Finland, 2004. [Set a(n)=d(n+3) on p. 8]
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,0,1,-3,3,-1).
%F a(n) = degree(p(n)) with p(n) = (x^(n-1)*p(n-1)*p(n-8) + p(n-4)*p(n-5))/p(n-9).
%F From _Colin Barker_, Oct 08 2019: (Start)
%F G.f.: x^2 / ((1-x)^4*(1+x)*(1+x^2)*(1+x^4)).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-8) - 3*a(n-9) + 3*a(n-10) - a(n-11) for n > 10. (End)
%F a(n) = (1/192)*(4*n^3 +42*n^2 +80*n -63 +3*(-1)^n) + (1/32)*(i^n*(1 + (-1)^n) + i^(n+1)*(1-(-1)^n)) + (1/4)*(b(n) -b(n-1) -2*b(n-2) -2*b(n -3)), where b(n) = A014017(n). - _G. C. Greubel_, Dec 29 2022
%t p[n_]:= p[n] = If[n<0, 1, Cancel[Simplify[(x^(n-1)*p[n-1]*p[n-8] + p[n-4]*p[n-5])/p[n-9]]]]; Table[Exponent[p[n], x], {n,0,30}]
%t LinearRecurrence[{3,-3,1,0,0,0,0,1,-3,3,-1}, {0,0,1,3,6,10,15,21,28,36, 46,58,72}, 61] (* _G. C. Greubel_, Dec 29 2022 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 72); [0,0] cat Coefficients(R!( x^2/((1-x)^4*(1+x)*(1+x^2)*(1+x^4)) )); // _G. C. Greubel_, Dec 29 2022
%o (SageMath)
%o def A122056_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^2/((1-x)^4*(1+x)*(1+x^2)*(1+x^4)) ).list()
%o A122056_list(70) # _G. C. Greubel_, Dec 29 2022
%Y Cf. A006731, A014017, A014125.
%K nonn,easy
%O 0,4
%A _Roger L. Bagula_, Sep 13 2006
%E Edited by _G. C. Greubel_, Dec 29 2022
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