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A108674 a(n) = (n+1)^2 * (n+2)^2 * (2*n+3) / 12. 3
1, 15, 84, 300, 825, 1911, 3920, 7344, 12825, 21175, 33396, 50700, 74529, 106575, 148800, 203456, 273105, 360639, 469300, 602700, 764841, 960135, 1193424, 1470000, 1795625, 2176551, 2619540, 3131884, 3721425, 4396575, 5166336, 6040320, 7028769, 8142575 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Kekulé numbers for certain benzenoids.

This is the case P(3,n) of the family of sequences defined in A132458. - Ottavio D'Antona (dantona(AT)dico.unimi.it), Oct 31 2007

Using the triangular numbers 0, 1, 3, ..., create a sequence of advancing sums of k-tuples with k=n*(n+1)/2 of the odd numbers: 0, 1, 15, 84, 300, 825, 1911, 3920, ... . This begins 0, then 1, then 3+5+7=15, then 9+11+13+15+17+19=84, then 21+23+...+39=300 and so on. - J. M. Bergot, Dec 08 2014

Partial sums of A008354. - J. M. Bergot, Dec 19 2014

Coefficients in the terminating series identity 1 - 15*n/(n + 4) + 84*n*(n - 1)/((n + 4)*(n + 5)) - 300*n*(n - 1)*(n - 2)/((n + 4)*(n + 5)*(n + 6)) + ... = 0 for n = 2,3,4,.... Cf. A000330. - Peter Bala, Feb 12 2019

REFERENCES

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 231, # 33).

LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

FORMULA

G.f.: (1+z)*(1+8*z+z^2)/(1-z)^6.

Also a(n) = Sum[Sum[i*j^2, {i, 1, n+1}], {j, 1, n+1}]. - Alexander Adamchuk, Jun 25 2006

MAPLE

a:=n->(n+1)^2*(n+2)^2*(2*n+3)/12; seq(a(n), n=0..35);

a:=n->sum(k^2*sum(k, k=0..n+1), k=0..n+1): seq(a(n), n=0...35); # Zerinvary Lajos, Aug 01 2008

MATHEMATICA

Table[(n^2+4*n^3+5*n^4+2*n^5)/12, {n, 40}] (* Enrique Pérez Herrero, Feb 27 2013 *)

PROG

(PARI) a(n)=(n+1)^2*(n+2)^2*(2*n+3)/12 \\ Charles R Greathouse IV, Feb 27 2013

(GAP) List([0..40], k->(k+1)^2*(k+2)^2*(2*k+3)/12); # Muniru A Asiru, Feb 18 2019

CROSSREFS

Cf. A000537, A126274.

Cf. A000330, A008354.

Sequence in context: A270768 A252935 A247958 * A050405 A241220 A279740

Adjacent sequences:  A108671 A108672 A108673 * A108675 A108676 A108677

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Jun 17 2005

STATUS

approved

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Last modified April 6 11:51 EDT 2020. Contains 333273 sequences. (Running on oeis4.)