A108646 a(n) = (n+1)*(n+2)^2*(n+3)*(11*n^3 + 58*n^2 + 101*n + 60)/720.

1, 23, 194, 985, 3668, 11074, 28728, 66438, 140415, 276001, 511082, 900263, 1519882, 2473940, 3901024, 5982300, 8950653, 13101051, 18802210, 26509637, 36780128, 50287798, 67841720, 90405250, 119117115, 155314341, 200557098

Offset: 0

Author

Emeric Deutsch, Jun 13 2005

Keywords

• nonn
• easy

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. 230, no. 22).

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A108645, A108647, A108648, A108649, A108650.

Programs

Magma

[(n+2)*(11*n^3+58*n^2+101*n+60)*Binomial(n+3,3)/120: n in [0..40]]; // G. C. Greubel, Oct 19 2023

Maple

a:=(n+1)*(n+2)^2*(n+3)*(11*n^3+58*n^2+101*n+60)/720: seq(a(n),n=0..30);

Mathematica

Table[(n+2)*(n+3)!*(11*n^3+58*n^2+101*n+60)/(6!*n!), {n,0,40}] (* G. C. Greubel, Oct 19 2023 *)

Python

A108646_list, m = [], [77, -85, 28, -1, 1, 1, 1, 1]
for _ in range(10001):
    A108646_list.append(m[-1])
    for i in range(7):
        m[i+1] += m[i] # Chai Wah Wu, Jun 12 2016

SageMath

[(n+2)*(11*n^3+58*n^2+101*n+60)*binomial(n+3,3)/120 for n in range(41)] # G. C. Greubel, Oct 19 2023

Formula

From Chai Wah Wu, Jun 12 2016: (Start)

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n > 7.

G.f.: (1 + 15*x + 38*x^2 + 21*x^3 + 2*x^4)/(1 - x)^8. (End)

E.g.f.: (1/6!)*(720 + 15840*x + 53640*x^2 + 56520*x^3 + 24030*x^4 + 4548*x^5 + 377*x^6 + 11*x^7)*exp(x). - G. C. Greubel, Oct 19 2023