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
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 12 2018
STATUS
approved