A304505 a(n) = 4*(n+1)*(9*n+4).

16, 104, 264, 496, 800, 1176, 1624, 2144, 2736, 3400, 4136, 4944, 5824, 6776, 7800, 8896, 10064, 11304, 12616, 14000, 15456, 16984, 18584, 20256, 22000, 23816, 25704, 27664, 29696, 31800, 33976, 36224, 38544, 40936, 43400, 45936, 48544, 51224, 53976, 56800, 59696

Offset: 0

Author

Emeric Deutsch, May 14 2018

Keywords

• nonn
• easy

Comments

a(n) is the first Zagreb index of the single-defect 4-gonal nanocone CNC(4,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(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.

9*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).

Crossrefs

Cf. A008586, A017209, A062708, A304506.

Programs

GAP

List([0..50],n->4*(n+1)*(9*n+4)); # Muniru A Asiru, May 14 2018

Maple

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

PARI

a(n) = 4*(n+1)*(9*n+4); \\ Altug Alkan, May 14 2018

PARI

Vec(8*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

Formula

From Colin Barker, May 14 2018: (Start)

G.f.: 8*(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.: 4*exp(x)*(4 + 22*x + 9*x^2).

a(n) = 8*A062708(n+1) = A017209(n)*A008586(n+1). (End)