A110690 Kekulé numbers for certain benzenoids.

1, 22, 193, 1045, 4180, 13566, 37764, 93456, 210705, 440440, 864721, 1610401, 2866864, 4908580, 8123280, 13046616, 20404233, 31162242, 46587145, 68316325, 98440276, 139597810, 195085540, 268983000, 366294825, 493111476, 656790057

Offset

0,2

References

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 243, H*(2,6,n)).

Links

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

a(n) = (n+1)*(n+2)^2*(n+3)*(n+4)(31*n^3 + 236*n^2 + 545*n + 420)/20160.

G.f.: (1 + 13*x + 31*x^2 + 16*x^3 + x^4)/(1-x)^9. - R. J. Mathar, Nov 01 2015

Maple

a:=n->(n+1)*(n+2)^2*(n+3)*(n+4)*(31*n^3+236*n^2+545*n+420)/20160: seq(a(n), n=0..31);

Mathematica

CoefficientList[Series[(1+13*x+31*x^2+16*x^3+x^4)/(1-x)^9, {x, 0, 50}], x] (* G. C. Greubel, Sep 06 2017 *)
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 22, 193, 1045, 4180, 13566, 37764, 93456, 210705}, 30] (* Harvey P. Dale, Nov 05 2019 *)

Prog

(Python)
A110690_list, m = [], [62, -65, 20, 0, 1, 1, 1, 1, 1]
for _ in range(10001):
    A110690_list.append(m[-1])
    for i in range(8):
        m[i+1] += m[i] # Chai Wah Wu, Jun 12 2016
(PARI) x='x+O('x^50); Vec((1+13*x+31*x^2+16*x^3+x^4)/(1-x)^9) \\ G. C. Greubel, Sep 06 2017

Crossrefs

Sequence in context: A022682 A107959 A200936 * A020923 A254470 A230896

Adjacent sequences: A110687 A110688 A110689 * A110691 A110692 A110693

Keyword

nonn,easy

Author

Emeric Deutsch, Aug 02 2005

Status

approved