A304618 a(n) = 108*n^2 - 228*n + 114 (n>=2).

90, 402, 930, 1674, 2634, 3810, 5202, 6810, 8634, 10674, 12930, 15402, 18090, 20994, 24114, 27450, 31002, 34770, 38754, 42954, 47370, 52002, 56850, 61914, 67194, 72690, 78402, 84330, 90474, 96834, 103410, 110202, 117210, 124434, 131874, 139530, 147402, 155490, 163794

Offset

2,1

Comments

For n>=2, a(n) is the first Zagreb index of the hexagonal network HX(n).

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 the hexagonal network HX(n) is M(HX(n); x,y) = 12*x^3*y^4 + 6*x^3*y^6 + 6*(n-3)*x^4*y^4 + 12*(n-2)*x^4*y^6 + (9*n^2-33*n+30)*x^6*y^6.

3*a(n) + 19 is a square. - Bruno Berselli, May 18 2018

Links

Colin Barker, Table of n, a(n) for n = 2..1000

E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.

S. Hayat and M. Imran, Computation of topological indices of certain networks, Applied Mathematics and Computation, 240, 2014, 213-228/

M. N. Husin and R. Hasni, More results on computation of topological indices of certain networks, Int. J. Networking and Virtual Organisations, 17, No. 1, 2017, 46-63.

B. Rajan, A. William, C. Grigorius, and S. Stephen, On certain topological indices of silicate, honeycomb and hexagonal networks, J. Comp. & Math. Sci., 3, No. 5, 2012, 530-535.

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

Formula

From Colin Barker, May 18 2018: (Start)

G.f.: 6*x^2*(15 + 22*x - x^2) / (1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.

(End)

Maple

seq(114-228*n+108*n^2, n = 2 .. 40);

Prog

(GAP) List([2..40], n->108*n^2-228*n+114); # Muniru A Asiru, May 18 2018
(PARI) a(n) = 108*n^2 - 228*n + 114; \\ Altug Alkan, May 18 2018
(PARI) Vec(6*x^2*(15 + 22*x - x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 18 2018

Crossrefs

Cf. A304619.

Sequence in context: A234991 A234984 A158490 * A187300 A237129 A250870

Adjacent sequences: A304615 A304616 A304617 * A304619 A304620 A304621

Keyword

nonn,easy

Author

Emeric Deutsch, May 17 2018

Status

approved