A304616 a(n) = 81*n^2 - 69*n + 24.

24, 36, 210, 546, 1044, 1704, 2526, 3510, 4656, 5964, 7434, 9066, 10860, 12816, 14934, 17214, 19656, 22260, 25026, 27954, 31044, 34296, 37710, 41286, 45024, 48924, 52986, 57210, 61596, 66144, 70854, 75726, 80760, 85956, 91314, 96834, 102516, 108360, 114366, 120534, 126864

Offset

0,1

Comments

For n>=1, a(n) = the first Zagreb index of the triangular silicate network TSL(n), defined pictorially in the Rosary et al. reference.

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 TSL(n) is M(TSL(n); x,y) = 3*x^3*y^3 + 3*(3*n-1)*x^3*y^7 + 3*(n-1)*(n-2)*x^3*y^12 + 3*(n-1)*x^7*y^7 + 6(n-2)*x^7*y^12 + 3*(n-2)*(n-3)*x^12*y^12/2.

36*a(n) - 335 is a square. - Bruno Berselli, May 18 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.

M. Rosary, C. J. Deeni, D. Antony Xavier, Computing some topological indices of triangular silicate network, Proceedings of the International Conference on Applied Mathematics and Theoretical Computer Science, 2013.

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

Formula

From Colin Barker, May 18 2018: (Start)

G.f.: 6*(4 - 6*x + 29*x^2) / (1 - x)^3.

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

(End)

Maple

seq(81*n^2-69*n+24, n = 0 .. 40);

Mathematica

Table[81n^2-69n+24, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {24, 36, 210}, 50] (* Harvey P. Dale, Feb 03 2021 *)

Prog

(GAP) List([0..40], n->81*n^2-69*n+24); # Muniru A Asiru, May 17 2018
(PARI) a(n) = 81*n^2 - 69*n + 24; \\ Altug Alkan, May 18 2018
(PARI) Vec(6*(4 - 6*x + 29*x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 18 2018

Crossrefs

Cf. A304617.

Sequence in context: A108190 A185489 A303884 * A172424 A241286 A334803

Adjacent sequences: A304613 A304614 A304615 * A304617 A304618 A304619

Keyword

nonn,easy

Author

Emeric Deutsch, May 17 2018

Status

approved