A304508 a(n) = 5*(3*n+1)*(9*n+8)/2 (n>=0).

20, 170, 455, 875, 1430, 2120, 2945, 3905, 5000, 6230, 7595, 9095, 10730, 12500, 14405, 16445, 18620, 20930, 23375, 25955, 28670, 31520, 34505, 37625, 40880, 44270, 47795, 51455, 55250, 59180, 63245, 67445, 71780, 76250, 80855, 85595, 90470, 95480, 100625, 105905, 111320

Offset: 0

Author

Emeric Deutsch, May 14 2018

Keywords

• nonn
• easy

Comments

The second Zagreb index of the single-defect 5-gonal nanocone CNC(5,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(5,n) is M(CNC(5,n); x,y) = 5*x^2*y^2 + 10*n*x^2*y^3 + 5*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.

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. A016777, A017257, A304507.

Programs

Maple

seq((1/2)*(5*(3*n+1))*(9*n+8), n = 0 .. 40);

Mathematica

Array[5 (3 # + 1) (9 # + 8)/2 &, 41, 0] (* or *)
LinearRecurrence[{3, -3, 1}, {20, 170, 455}, 41] (* or *)
CoefficientList[Series[5 (4 + 22 x + x^2)/(1 - x)^3, {x, 0, 40}], x] (* Michael De Vlieger, May 14 2018 *)

PARI

a(n) = 5*(3*n+1)*(9*n+8)/2; \\ Altug Alkan, May 14 2018

PARI

Vec(5*(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.: 5*(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.: 5*exp(x)*(8 + 60*x + 27*x^2)/2.

a(n) = 5*A016777(n)*A017257(n)/2. (End)