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A078876
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a(n) = n^4*(n^4-1)/240.
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1
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0, 0, 1, 27, 272, 1625, 6993, 24010, 69888, 179334, 416625, 893101, 1791504, 3398759, 6148961, 10678500, 17895424, 29065308, 45916065, 70764303, 106666000, 157594437, 228648497, 326294606, 458645760, 635781250, 870110865, 1176787521, 1574172432, 2084357107
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OFFSET
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0,4
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COMMENTS
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For n>=2, the triple (n^6, 120*a(n), (n^8 + n^4)/2) form a Pythagorean triple whose short leg is a square and the other sides are triangular numbers. - Michel Marcus, Mar 15 2021
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REFERENCES
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S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, #14).
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LINKS
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FORMULA
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G.f.: -x^2*(x+1)*(x^4+17*x^3+48*x^2+17*x+1) / (x-1)^9. - Colin Barker, Jun 18 2013
Sum_{n>=2} 1/a(n) = 450 - 8*Pi^4/3 - 60*Pi*coth(Pi).
Sum_{n>=2} (-1)^n/a(n) = 7*Pi^4/3 - 60*Pi*cosech(Pi) - 210. (End)
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MATHEMATICA
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Table[n^4*(n^4 - 1)/240, {n, 0, 30}] (* Amiram Eldar, May 31 2022 *)
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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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STATUS
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approved
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