A078876 - OEIS

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A078876

a(n) = n^4*(n^4-1)/240.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005

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

REFERENCES

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, #14).

LINKS

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

FORMULA

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

From Amiram Eldar, May 31 2022: (Start)

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)

MATHEMATICA

Table[n^4*(n^4 - 1)/240, {n, 0, 30}] (* Amiram Eldar, May 31 2022 *)