A304374 a(n) = 9*n^2 + 21*n - 6 (n>=1).

24, 72, 138, 222, 324, 444, 582, 738, 912, 1104, 1314, 1542, 1788, 2052, 2334, 2634, 2952, 3288, 3642, 4014, 4404, 4812, 5238, 5682, 6144, 6624, 7122, 7638, 8172, 8724, 9294, 9882, 10488, 11112, 11754, 12414, 13092, 13788, 14502, 15234, 15984, 16752, 17538, 18342, 19164, 20004, 20862, 21738, 22632, 23544

Offset

1,1

Comments

a(n) is the first Zagreb index of the triangular benzenoid T(n) (see the M. Ghorbani et al. references).

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 triangular benzenoid T(n) is M(T(n); x,y) = 6*x^2*y^2 + 6*(n-1)*x^2*y^3 + 3*n*(n-1)*x^3*y^3/2.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

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

M. Ghorbani and M. Ghazi, Computing geometric arithmetic index of some graphs, Digest J. of Nanomaterials and Biostructures, 5, No. 4, 2010, 837-841.

M. Ghorbani and M. Ghazi, Computing some topological indices of triangular benzenoid, Digest J. of Nanomaterials and Biostructures, 5, No. 4, 2010, 1107-1111.

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

Formula

From Colin Barker, May 12 2018: (Start)

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

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

(End)

Maple

seq(9*n^2 + 21*n - 6, n=1..50);

Mathematica

Table[9n^2+21n-6, {n, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {24, 72, 138}, 50] (* Harvey P. Dale, Apr 11 2024 *)

Prog

(PARI) Vec(6*x*(2 - x)*(2 + x) /(1 - x)^3 + O(x^40)) \\ Colin Barker, May 12 2018
(PARI) a(n) = 9*n^2 + 21*n - 6; \\ Altug Alkan, May 12 2018

Crossrefs

Cf. A304375.

Sequence in context: A143337 A183006 A090860 * A064200 A305065 A192833

Adjacent sequences: A304371 A304372 A304373 * A304375 A304376 A304377

Keyword

nonn,easy,changed

Author

Emeric Deutsch, May 12 2018

Status

approved