A109114 a(n) = 5*a(n-1) - 3*a(n-2), a(0)=1, a(1)=6.

1, 6, 27, 117, 504, 2169, 9333, 40158, 172791, 743481, 3199032, 13764717, 59226489, 254838294, 1096512003, 4718045133, 20300689656, 87349312881, 375844495437, 1617174538542, 6958339206399, 29940172416369, 128825844462648

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. 302, P_{11}).

Links

Table of n, a(n) for n=0..22.

Index entries for linear recurrences with constant coefficients, signature (5,-3).

Formula

a(n) = ((sqrt(13) + 7)*((5 + sqrt(13))/2)^n + (sqrt(13) - 7)*((5 - sqrt(13))/2)^n)/(2*sqrt(13)).

G.f.: (1+z)/(1 - 5z + 3z^2).

a(n) = A116415(n)+A116415(n-1). - R. J. Mathar, Jul 26 2022

Maple

a[0]:=1: a[1]:=6: for n from 2 to 26 do a[n]:=5*a[n-1]-3*a[n-2] od: seq(a[n], n=0..26);

Mathematica

LinearRecurrence[{5, -3}, {1, 6}, 30] (* Harvey P. Dale, Dec 03 2012 *)

Prog

(PARI) a(n)=([0, 1; -3, 5]^n*[1; 6])[1, 1] \\ Charles R Greathouse IV, May 27 2026

Crossrefs

Cf. A116415.

Sequence in context: A387604 A196919 A049651 * A373302 A080619 A265943

Adjacent sequences: A109111 A109112 A109113 * A109115 A109116 A109117

Keyword

nonn,easy,changed

Author

Emeric Deutsch, Jun 19 2005

Status

approved