A304504 -id:A304504 - cp's OEIS frontend

12, 78, 198, 372, 600, 882, 1218, 1608, 2052, 2550, 3102, 3708, 4368, 5082, 5850, 6672, 7548, 8478, 9462, 10500, 11592, 12738, 13938, 15192, 16500, 17862, 19278, 20748, 22272, 23850, 25482, 27168, 28908, 30702, 32550, 34452, 36408, 38418, 40482, 42600, 44772

Offset: 0

Emeric Deutsch, May 13 2018

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

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).

Cf. A008585, A017209, A062708, A304504.

Maple
```
seq((3*(n+1))*(9*n+4), n = 0 .. 40);
```

Vec(6*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

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).
a(n) = 6*A062708(n+1) = A017209(n)*A008585(n+1). (End)

cp's OEIS Frontend

A304503 a(n) = 3(n+1)(9*n+4).

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