

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 ktuples 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)/(1z)^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



