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A108678 a(n) = (n+1)^2*(n+2)*(2*n+3)/6. 5
1, 10, 42, 120, 275, 546, 980, 1632, 2565, 3850, 5566, 7800, 10647, 14210, 18600, 23936, 30345, 37962, 46930, 57400, 69531, 83490, 99452, 117600, 138125, 161226, 187110, 215992, 248095, 283650, 322896, 366080, 413457, 465290, 521850, 583416, 650275, 722722 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Kekulé numbers for certain benzenoids.
REFERENCES
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 232, # 44).
LINKS
FORMULA
G.f.: (1 + 5*x + 2*x^2)/(1-x)^5.
a(n) = A098077(n+1)/2. - Alexander Adamchuk, Apr 12 2006
From Amiram Eldar, May 31 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi^2 + 48*log(2) - 42.
Sum_{n>=0} (-1)^n/a(n) = Pi^2/2 - 12*Pi - 12*log(2) + 42. (End)
From G. C. Greubel, Apr 09 2023: (Start)
a(n) = (1/3)*binomial(n+2, 2)*binomial(2*n+3, 2).
a(n) = (1/3)*A000217(n+1)*A014105(n+1)
a(n) = (1/8)*A100431(n).
E.g.f.: (1/6)*(6 + 54*x + 69*x^2 + 23*x^3 + 2*x^4)*exp(x). (End)
a(n) = (n+1)*A000330(n+1). - Olivier Gérard, Jan 13 2024
MAPLE
a:=n->(n+1)^2*(n+2)*(2*n+3)/6: seq(a(n), n=0..42);
a:=n->sum(n*j^2, j=1..n): seq(a(n), n=1..36); # Zerinvary Lajos, Apr 29 2007
MATHEMATICA
Table[(n+1)^2*(n+2)(2n+3)/6, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)
PROG
(Magma) [(n+1)^2*(n+2)*(2*n+3)/6: n in [0..60]]; // G. C. Greubel, Apr 09 2023
(SageMath) [(n+1)^2*(n+2)*(2*n+3)/6 for n in range(61)] # G. C. Greubel, Apr 09 2023
CROSSREFS
Sequence in context: A372666 A328536 A163815 * A226988 A358249 A027171
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 17 2005
STATUS
approved

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Last modified May 23 05:47 EDT 2024. Contains 372758 sequences. (Running on oeis4.)