A304506 a(n) = 2*(3*n+1)*(9*n+8).

16, 136, 364, 700, 1144, 1696, 2356, 3124, 4000, 4984, 6076, 7276, 8584, 10000, 11524, 13156, 14896, 16744, 18700, 20764, 22936, 25216, 27604, 30100, 32704, 35416, 38236, 41164, 44200, 47344, 50596, 53956, 57424, 61000, 64684, 68476, 72376, 76384, 80500, 84724, 89056

Offset: 0

Author

Emeric Deutsch, May 14 2018

Keywords

• nonn
• easy

Comments

a(n) is the second Zagreb index of the single-defect 4-gonal nanocone CNC(4,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(4,n) is M(CNC(4,n);x,y) = 4*x^2*y^2 + 8*n*x^2*y^3 + 2*n*(3*n+1)*x^3*y^3.

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.

6*a(n) + 25 is a square. - Bruno Berselli, May 14 2018

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• 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.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A016933, A017257, A304505.

Programs

GAP

List([0..50],n->2*(3*n+1)*(9*n+8)); # Muniru A Asiru, May 14 2018

Maple

seq((2*(9*n+8))*(3*n+1), n = 0 .. 40);

Mathematica

Table[2(3n+1)(9n+8),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{16,136,364},50] (* Harvey P. Dale, Aug 15 2022 *)

PARI

a(n) = 2*(3*n+1)*(9*n+8); \\ Altug Alkan, May 14 2018

PARI

Vec(4*(4 + 22*x + x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

Formula

From Colin Barker, May 14 2018: (Start)

G.f.: 4*(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.: 2*exp(x)*(8 + 60*x + 27*x^2).

a(n) = A016933(n)*A017257(n). (End)