OFFSET
0,1
COMMENTS
The first Zagreb index of the single-defect 3-gonal nanocone CNC(3,n) (see definition in the Doslic et al. reference, p. 27).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums 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.
12*a(n) + 25 is a square. - Bruno Berselli, May 14 2018
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, J. 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.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From Colin Barker, May 14 2018: (Start)
G.f.: 6*(2 + 7*x)/(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)*(4 + 22*x + 9*x^2).
MAPLE
seq((3*(n+1))*(9*n+4), n = 0 .. 40);
MATHEMATICA
PROG
(PARI) Vec(6*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Emeric Deutsch, May 13 2018
STATUS
approved