A304157 a(n) is the first Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference.

24, 68, 112, 156, 200, 244, 288, 332, 376, 420, 464, 508, 552, 596, 640, 684, 728, 772, 816, 860, 904, 948, 992, 1036, 1080, 1124, 1168, 1212, 1256, 1300, 1344, 1388, 1432, 1476, 1520, 1564, 1608, 1652, 1696, 1740, 1784, 1828, 1872, 1916, 1960, 2004, 2048

Offset: 1

Author

Emeric Deutsch, May 07 2018

Keywords

• nonn
• easy

Comments

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 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.

a(n) is the first Zagreb index of the angular phenylene shown in the Bodroza-Pantic et al. reference (Fig. 1 (b)). - Emeric Deutsch, May 24 2018

Examples

			From _Andrew Howroyd_, May 09 2018: (Start)
Illustration of the first two graphs:
       o              o         o
     /   \          /   \     /   \
    o     o        o     o---o     o
    |     |        |     |   |     |
    o     o        o     o---o     o
     \   /          \   /     \   /
       o              o         o
In general, the graph consists of a chain of n linked hexagons.
.
Case n=1: There are 6 vertices of degree 2, so a(1) = 6*2^2 = 24.
Case n=2: There are 8 vertices of degree 2 and 4 of degree 3, so a(2) = 8*2^2 + 4*3^3 = 32 + 36 = 68.
In general, there will be 2n + 4 vertices of degree 2 and 4n - 4 of degree 3.
(End)
		

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).

Crossrefs

Cf. A017461, A224454, A224455, A304158.

Programs

Maple

seq(44*n - 20, n = 1 .. 40);

PARI

Vec(4*x*(6 + 5*x) / (1 - x)^2 + O(x^60)) \\ Colin Barker, May 07 2018

Formula

a(n) = 44*n - 20.

a(n) = 4 * A017461(n-1).

From Colin Barker, May 07 2018: (Start)

G.f.: 4*x*(6 + 5*x) / (1 - x)^2.

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

(End)