login
A304504
a(n) = 3*(3*n+1)*(9*n+8)/2.
2
12, 102, 273, 525, 858, 1272, 1767, 2343, 3000, 3738, 4557, 5457, 6438, 7500, 8643, 9867, 11172, 12558, 14025, 15573, 17202, 18912, 20703, 22575, 24528, 26562, 28677, 30873, 33150, 35508, 37947, 40467, 43068, 45750, 48513, 51357, 54282, 57288, 60375, 63543, 66792
OFFSET
0,1
COMMENTS
The second Zagreb index of the single-defect 3-gonal nanocone CNC(3,n) (see definition in the Doslic et al. reference, p. 27).
The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.
The M-polynomial of CNC(3,n) is M(CNC(3,n); x,y) = 3*x^2*y^2 + 6*n*x^2*y^3 + 3*n*(3*n+1)*x^3*y^3/2.
More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n); x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2.
8*a(n) + 25 is a square. - Bruno Berselli, May 14 2018
LINKS
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian Journal of Mathematical Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, Journal of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.
A. Khaksar, M. Ghorbani, and H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials - Rapid Communications, Vol. 4, No. 11, 2010, pp. 1868-1870.
FORMULA
From Colin Barker, May 14 2018: (Start)
G.f.: 3*(4 + 22*x + x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: 3*exp(x)*(8 + 60*x + 27*x^2)/2.
a(n) = A017197(n)*A017257(n)/2. (End)
MAPLE
seq((1/2)*(3*(9*n+8))*(3*n+1), n = 0 .. 40);
PROG
(PARI) Vec(3*(4 + 22*x + x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Emeric Deutsch, May 13 2018
STATUS
approved