A304158 - OEIS

A304158

a(n) is the second Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference (Fig. 3).

24, 84, 144, 204, 264, 324, 384, 444, 504, 564, 624, 684, 744, 804, 864, 924, 984, 1044, 1104, 1164, 1224, 1284, 1344, 1404, 1464, 1524, 1584, 1644, 1704, 1764, 1824, 1884, 1944, 2004, 2064, 2124, 2184, 2244, 2304, 2364, 2424, 2484, 2544, 2604, 2664, 2724, 2784, 2844, 2904, 2964

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OFFSET

1,1

COMMENTS

The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.

The M-polynomial of the linear phenylene G[n] is M(G[n];x,y) = 6*x^2*y^2 + 4*(n - 1)*x^2*y^3 + 4(n - 1)*x^3*y^3.

LINKS

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

O. Bodroza-Pantic, I. Gutman, and S. J. Cyvin, Fibonacci numbers and algebraic structure count of some non-benzenoid conjugated polymers, The Fibonacci Quarterly, 35, 1, 1997, 75-83.

M. R. Darafsheh, Computation of topological indices of some graphs, Acta Appl. Math., 110, 2010, 1225-1235.

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

P. Gayathri and U. Priyanka, Degree based topological indices of linear phenylene, Internat. J. of Innovative Research in Science, Engineering and Technology, 6, 8, 2017, 16986-16997.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

a(n) = 60*n - 36.

a(n) = 12 * A016873(n-1). - Alois P. Heinz, May 09 2018

From Bruno Berselli, May 09 2018: (Start)

O.g.f.: 12*x*(2 + 3*x)/(1 - x)^2.

E.g.f.: 12*(3 - 3*exp(x) + 5*x*exp(x)).

a(n) = 2*a(n-1) - a(n-2).

a(n) = A008594(5*n-3) = A017317(6*n-4) = A072710(4*n-2) = A139245(3*n-1). (End)

EXAMPLE

a(1) = 24; indeed, G[1] is a hexagon; we have 6 edges, each with end vertices of degree 2; then the second Zagreb index is 6*2*2 =24.

MAPLE