

A304505


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


2



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
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OFFSET

0,1


COMMENTS

a(n) is the first Zagreb index of the singledefect 4gonal 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 Mpolynomial 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 Mpolynomial 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
E. Deutsch and Sandi Klavzar, Mpolynomial and degreebased topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of singledefect nanocones, J. of Mathematical Nanoscience, 1, No. 1, 2011, 2531.
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, 18681870.
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

From Colin Barker, May 14 2018: (Start)
G.f.: 8*(2 + 7*x) / (1  x)^3.
a(n) = 3*a(n1)  3*a(n2) + a(n3) for n>2.
(End)


MAPLE

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


PROG

(PARI) a(n) = 4*(n+1)*(9*n+4); \\ Altug Alkan, May 14 2018
(GAP) List([0..50], n>4*(n+1)*(9*n+4)); # Muniru A Asiru, May 14 2018
(PARI) Vec(8*(2 + 7*x) / (1  x)^3 + O(x^40)) \\ Colin Barker, May 14 2018


CROSSREFS

Cf. A304506.
Sequence in context: A240257 A199773 A245872 * A210687 A000739 A186852
Adjacent sequences: A304502 A304503 A304504 * A304506 A304507 A304508


KEYWORD

nonn,easy


AUTHOR

Emeric Deutsch, May 14 2018


STATUS

approved



