A304507 a(n) = 5*(n+1)*(9*n+4).

20, 130, 330, 620, 1000, 1470, 2030, 2680, 3420, 4250, 5170, 6180, 7280, 8470, 9750, 11120, 12580, 14130, 15770, 17500, 19320, 21230, 23230, 25320, 27500, 29770, 32130, 34580, 37120, 39750, 42470, 45280, 48180, 51170, 54250, 57420, 60680, 64030, 67470, 71000, 74620

Offset: 0

Author

Emeric Deutsch, May 14 2018

Keywords

• nonn
• easy

Comments

The first Zagreb index of the single-defect 5-gonal nanocone CNC(5,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(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, 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. A008587, A011862, A017209, A062708, A108579, A304508.

Programs

GAP

List([0..50], n -> 5*(n+1)*(9*n+4)); # Muniru A Asiru, May 15 2018

Maple

seq((5*(n+1))*(9*n+4), n = 0 .. 40);

Mathematica

Array[5 (# + 1) (9 # + 4) &, 41, 0] (* or *)
LinearRecurrence[{3, -3, 1}, {20, 130, 330}, 41] (* or *)
CoefficientList[Series[10 (2 + 7 x)/(1 - x)^3, {x, 0, 40}], x] (* Michael De Vlieger, May 14 2018 *)

PARI

a(n) = 5*(n+1)*(9*n+4); \\ Altug Alkan, May 14 2018

PARI

Vec(10*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

Formula

From Colin Barker, May 14 2018: (Start)

G.f.: 10*(2 + 7*x)/(1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

a(n) = 10*A062708(n+1) for n >= 0. - Robert G. Wilson v, May 14 2018

a(n) = 5*A011862(9*n+7) = 5*A108579(6*n+7). - Bruno Berselli, May 15 2018

From Elmo R. Oliveira, Nov 15 2024: (Start)

E.g.f.: 5*exp(x)*(4 + 22*x + 9*x^2).

a(n) = 5*A017209(n)*A008587(n+1). (End)