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
KEYWORD
nonn,easy,changed
AUTHOR
Emeric Deutsch, May 17 2018
STATUS
approved