A123350 a(n) = (n^4 + 2n^3 + 5n^2 + 4)/4.

1, 3, 14, 46, 117, 251, 478, 834, 1361, 2107, 3126, 4478, 6229, 8451, 11222, 14626, 18753, 23699, 29566, 36462, 44501, 53803, 64494, 76706, 90577, 106251, 123878, 143614, 165621, 190067, 217126, 246978, 279809, 315811, 355182, 398126, 444853

Offset

0,2

Comments

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Oct 16 2006

Form the 2 X 3 matrix with first row C(n,0), C(n,1), and C(n,2) and second row C(n+1,0), C(n+1,1), and C(n+1,2), multiply it by its transpose to get a 2 X 2 matrix: its determinant = a(n). - J. M. Bergot, Sep 05 2013

References

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

G.f.: (-1 + 2*x - 9*x^2 + 4*x^3 - 2*x^4) / (x-1)^5 . - R. J. Mathar, Oct 19 2012

a(n) = 1 + A117717(n+1). - R. J. Mathar, Sep 15 2013

E.g.f.: (x^4 + 8*x^3 + 18*x^2 + 8*x + 4)*exp(x)/4. - G. C. Greubel, Oct 12 2017

Maple

a:=n->(n^4+2*n^3+5*n^2+4)/4: seq(a(n), n=0..40); # Emeric Deutsch, Oct 16 2006

Mathematica

Table[(n^4 + 2*n^3 + 5*n^2 + 4)/4, {n, 0, 50}] (* G. C. Greubel, Oct 12 2017 *)

Prog

(PARI) for(n=0, 50, print1((n^4 + 2*n^3 + 5*n^2 + 4)/4, ", ")) \\ G. C. Greubel, Oct 12 2017
(Magma) [(n^4 + 2*n^3 + 5*n^2 + 4)/4: n in [0..30]]; // G. C. Greubel, Oct 12 2017

Crossrefs

Sequence in context: A032150 A032055 A180785 * A027953 A264501 A104196

Adjacent sequences: A123347 A123348 A123349 * A123351 A123352 A123353

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 10 2006

Extensions

More terms from Emeric Deutsch, Oct 16 2006

Status

approved