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.

Number of rectangles in an n+2 X n+2 grid of points with sides parallel or at 45-degree angles to the axes. - David Radcliffe, May 24 2026

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

Dmitry Babichev and Sergey Babichev, Counting All Lattice Rectangles in the Square Grid in Near-Linear Time, arXiv:2604.22456 [cs.CG], 2026.

Andrii Husiev, Extended Central Factorial Numbers and the Flickering Operator, arXiv:2605.06689 [math.GM], 2026. See p. 5.

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

a(n) = A000537(n+1) + A002415(n+1). - David Radcliffe, May 24 2026

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