A122056 Expansion of g.f. x^2/((1 - x)^4*(1 + x)*(1 + x^2)*(1 + x^4)).

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 46, 58, 72, 88, 106, 126, 148, 172, 199, 229, 262, 298, 337, 379, 424, 472, 524, 580, 640, 704, 772, 844, 920, 1000, 1085, 1175, 1270, 1370, 1475, 1585, 1700, 1820, 1946, 2078, 2216, 2360, 2510, 2666, 2828, 2996, 3171, 3353, 3542, 3738

Offset: 0

Author

Roger L. Bagula, Sep 13 2006

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, arXiv:0807.2538 [nlin.SI], 2008; Proceedings of SIDE 6, Helsinki, Finland, 2004. [Set a(n)=d(n+3) on p. 8]
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,1,-3,3,-1).

Crossrefs

Cf. A006731, A014017, A014125.

Programs

Magma

R:=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

Mathematica

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}]
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 *)

SageMath

def A122056_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2/((1-x)^4*(1+x)*(1+x^2)*(1+x^4)) ).list()
A122056_list(70) # G. C. Greubel, Dec 29 2022

Formula

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).

From Colin Barker, Oct 08 2019: (Start)

G.f.: x^2 / ((1-x)^4*(1+x)*(1+x^2)*(1+x^4)).

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)

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

Extensions

Edited by G. C. Greubel, Dec 29 2022