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

Author

N. J. A. Sloane, Oct 10 2006

Keywords

• nonn
• easy

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).

Programs

Magma

[(n^4 + 2*n^3 + 5*n^2 + 4)/4: n in [0..30]]; // 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 *)

PARI

for(n=0,50, print1((n^4 + 2*n^3 + 5*n^2 + 4)/4, ", ")) \\ G. C. Greubel, Oct 12 2017

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

Extensions

More terms from Emeric Deutsch, Oct 16 2006