OFFSET
0,1
COMMENTS
a(n) is the first Zagreb index of the dendrimer nanostar G[n], defined pictorially in the Iranmanesh et al. reference (Fig. 1, where G[3] is shown) or in Alikhani et al. reference (Fig. 1, where G[3] is shown).
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 G[n] is M(G[n]; x, y) = 3*2^n*x*y^3 + (12*2^n - 6)*x^2*y^2 + (24*2^n -12)*x^2*y^3 + (9*2^n - 6)*x^3*y^3.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
S. Alikhani, M. A. Iranmanesh, Eccentric connectivity polynomials of an infinite family of dendrimer, Digest J. Nanomaterials and Biostructures, 6 (2011) 253-257.
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
A. Iranmanesh and N. Dorosti, Computing the Cluj index of a type dendrimer nanostars, MATCH Commun. Math. Comput. Chem. 65, 2011, 209-219.
Index entries for linear recurrences with constant coefficients, signature (3,-2).
FORMULA
From Colin Barker, May 25 2018: (Start)
G.f.: 6*(19 + x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2)for n>0. (End)
MAPLE
seq(234*2^n-120, n = 0 .. 40);
MATHEMATICA
234*2^Range[0, 35] - 120 (* Paolo Xausa, Jun 08 2026 *)
PROG
(PARI) Vec(6*(19 + x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 25 2018
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Emeric Deutsch, May 25 2018
STATUS
approved