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A304503 a(n) = 3*(n+1)*(9*n+4). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

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 J. Math. Chemistry, 6, No. 2, 2015, 93-102.

T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, J. of Mathematical Nanoscience, 1, No. 1, 2011, 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, 4, No. 11, 2010, 1868-1870.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

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>3.

(End)

MAPLE

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

PROG

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

CROSSREFS

Cf. A304504.

Sequence in context: A244390 A136540 A139612 * A268793 A162629 A008504

Adjacent sequences:  A304500 A304501 A304502 * A304504 A304505 A304506

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, May 13 2018

STATUS

approved

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)