A108649 a(n) = (n+1)*(n+2)*(n+3)*(13*n^3 + 69*n^2 + 113*n + 60)/360.

1, 17, 111, 457, 1428, 3710, 8442, 17382, 33099, 59191, 100529, 163527, 256438, 389676, 576164, 831708, 1175397, 1630029, 2222563, 2984597, 3952872, 5169802, 6684030, 8551010, 10833615, 13602771, 16938117, 20928691, 25673642, 31282968

Offset

0,2

Comments

Kekulé numbers for certain benzenoids.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 25).

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(0)=1, a(1)=17, a(2)=111, a(3)=457, a(4)=1428, a(5)=3710, a(6)=8442, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Harvey P. Dale, Jul 01 2012

G.f.: (1 + 10*x + 13*x^2 + 2*x^3) / (1 - x)^7. - Colin Barker, Apr 22 2020

E.g.f.: (1/360)*(360 + 5760*x + 14040*x^2 + 10440*x^3 + 2985*x^4 + 342*x^5 + 13*x^6)*exp(x). - G. C. Greubel, Oct 19 2023

Maple

a:=(n+1)*(n+2)*(n+3)*(13*n^3+69*n^2+113*n+60)/360: seq(a(n), n=0..36);

Mathematica

Table[(n+1)(n+2)(n+3)(13n^3+69n^2+113n+60)/360, {n, 0, 30}] (* or *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 17, 111, 457, 1428, 3710, 8442}, 30] (* Harvey P. Dale, Jul 01 2012 *)

Prog

(PARI) Vec((1+10*x+13*x^2+2*x^3)/(1-x)^7 + O(x^40)) \\ Colin Barker, Apr 22 2020
(Magma) [(13*n^3+69*n^2+113*n+60)*Binomial(n+3, 3)/60: n in [0..40]]; // G. C. Greubel, Oct 19 2023
(SageMath) [(13*n^3+69*n^2+113*n+60)*binomial(n+3, 3)/60 for n in range(41)] # G. C. Greubel, Oct 19 2023

Crossrefs

Cf. A108645, A108646, A108647, A108648, A108650.

Sequence in context: A261809 A264723 A157099 * A296260 A139858 A139903

Adjacent sequences: A108646 A108647 A108648 * A108650 A108651 A108652

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 13 2005

Status

approved