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A085409
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Sum of three solutions of the Diophantine equation x^2 - y^2 = z^3.
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5
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0, 12, 84, 270, 624, 1200, 2052, 3234, 4800, 6804, 9300, 12342, 15984, 20280, 25284, 31050, 37632, 45084, 53460, 62814, 73200, 84672, 97284, 111090, 126144, 142500, 160212, 179334, 199920, 222024, 245700, 271002, 297984, 326700, 357204, 389550, 423792, 459984
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OFFSET
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0,2
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COMMENTS
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Parametric representation of the solution is (x, y, z) = (6n^3, 3n^3, 3n^2), thus getting a(n) = 9n^3 + 3n^2.
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LINKS
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FORMULA
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a(n) = 9*n^3 + 3*n^2.
G.f.: 6*x*(2 + 6*x + x^2) /(1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
(End)
Sum_{n>=1} 1/a(n) = Pi^2/18 + sqrt(3)*Pi/6 + 3*log(3)/2 - 3.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36 - Pi/sqrt(3) - 2*log(2) + 3. (End)
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MATHEMATICA
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Table[9n^3 + 3n^2, {n, 0, 34}]
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PROG
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(PARI) concat(0, Vec(6*x*(2 + 6*x + x^2) /(1 - x)^4 + O(x^40))) \\ Colin Barker, Oct 25 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( 6*x*(2 + 6*x + x^2) /(1 - x)^4)); // Marius A. Burtea, Oct 25 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Aug 13 2003
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EXTENSIONS
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STATUS
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approved
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