A006731 -id:A006731 - cp's OEIS frontend

A056532 Bond percolation series for square lattice near a wall.

1, 1, 2, 3, 6, 9, 17, 26, 47, 72, 129, 194, 348, 516, 929, 1351, 2456, 3506, 6471, 8929, 17029, 22579, 44707, 55969, 117836, 137313, 311654, 324989, 833496, 756309, 2242031, 1623709, 6176873, 3240757, 17192674, 4663165, 49481888, 1180046, 144593684, -40561669, 439929287, -230303695, 1351358555, -1116634980, 4353263697

Offset: 0

N. J. A. Sloane, Aug 27 2000

sign

I. Jensen, Table of n, a(n) for n = 0..175 (from link below)
I. Jensen, More terms

Cf. A056575, A006731.

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

Original entry on oeis.org

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

Roger L. Bagula, Sep 13 2006

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).

Cf. A006731, A014017, A014125.

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

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 *)

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

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

Edited by G. C. Greubel, Dec 29 2022

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A056532 Bond percolation series for square lattice near a wall.

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

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cp's OEIS Frontend

A056532 Bond percolation series for square lattice near a wall.

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

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