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Bond percolation series for square lattice near a wall.
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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
Expansion of g.f. x^2/((1 - x)^4*(1 + x)*(1 + x^2)*(1 + x^4)).
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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
LINKS
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,1,-3,3,-1).
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).
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
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 *)
PROG
(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 (SageMath)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^2/((1-x)^4*(1+x)*(1+x^2)*(1+x^4)) ).list()
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