A108678 a(n) = (n+1)^2*(n+2)*(2*n+3)/6.

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

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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

Cf. A000217, A014105, A098077, A100431.

Sequence in context: A372666 A328536 A163815 * A226988 A358249 A027171

Adjacent sequences: A108675 A108676 A108677 * A108679 A108680 A108681

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 17 2005

Status

approved