

A304157


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


3



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
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OFFSET

1,1


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 Mpolynomial 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 BodrozaPantic et al. reference (Fig. 1 (b)).  Emeric Deutsch, May 24 2018


LINKS



FORMULA

a(n) = 44*n  20.
G.f.: 4*x*(6 + 5*x) / (1  x)^2.
a(n) = 2*a(n1)  a(n2) for n>2.
(End)


EXAMPLE

Illustration of the first two graphs:
o o o
/ \ / \ / \
o o o oo o
     
o o o oo 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)


MAPLE

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


PROG

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


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



