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

%I A304157 #35 Jan 05 2025 19:51:41
%S A304157 24,68,112,156,200,244,288,332,376,420,464,508,552,596,640,684,728,
%T A304157 772,816,860,904,948,992,1036,1080,1124,1168,1212,1256,1300,1344,1388,
%U A304157 1432,1476,1520,1564,1608,1652,1696,1740,1784,1828,1872,1916,1960,2004,2048
%N A304157 a(n) is the first Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference.
%C A304157 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.
%C A304157 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.
%C A304157 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
%H A304157 Colin Barker, <a href="/A304157/b304157.txt">Table of n, a(n) for n = 1..1000</a>
%H A304157 O. Bodroza-Pantic, I. Gutman, and S. J. Cyvin, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/35-1/bodroza-pantic.pdf">Fibonacci numbers and algebraic structure count of some non-benzenoid conjugated polymers</a>, The Fibonacci Quarterly, 35, 1, 1997, 75-83.
%H A304157 M. R. Darafsheh, <a href="https://doi.org/10.1007/s10440-009-9503-8">Computation of topological indices of some graphs</a>, Acta Appl. Math., 110, 2010, 1225-1235.
%H A304157 E. Deutsch and Sandi Klavzar, <a href="http://dx.doi.org/10.22052/ijmc.2015.10106">M-polynomial and degree-based topological indices</a>, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
%H A304157 P. Gayathri and U. Priyanka, <a href="https://www.ijirset.com/upload/2017/august/185_IJ60808680-%20final%20submission.pdf">Degree based topological indices of linear phenylene</a>, Internat. J. of  Innovative Research in Science, Engineering and Technology,6, 8, 2017, 16986-16997.
%H A304157 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A304157 a(n) = 44*n - 20.
%F A304157 a(n) = 4 * A017461(n-1).
%F A304157 From _Colin Barker_, May 07 2018: (Start)
%F A304157 G.f.: 4*x*(6 + 5*x) / (1 - x)^2.
%F A304157 a(n) = 2*a(n-1) - a(n-2) for n>2.
%F A304157 (End)
%e A304157 From _Andrew Howroyd_, May 09 2018: (Start)
%e A304157 Illustration of the first two graphs:
%e A304157        o              o         o
%e A304157      /   \          /   \     /   \
%e A304157     o     o        o     o---o     o
%e A304157     |     |        |     |   |     |
%e A304157     o     o        o     o---o     o
%e A304157      \   /          \   /     \   /
%e A304157        o              o         o
%e A304157 In general, the graph consists of a chain of n linked hexagons.
%e A304157 .
%e A304157 Case n=1: There are 6 vertices of degree 2, so a(1) = 6*2^2 = 24.
%e A304157 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.
%e A304157 In general, there will be 2n + 4 vertices of degree 2 and 4n - 4 of degree 3.
%e A304157 (End)
%p A304157 seq(44*n - 20, n = 1 .. 40);
%o A304157 (PARI) Vec(4*x*(6 + 5*x) / (1 - x)^2 + O(x^60)) \\ _Colin Barker_, May 07 2018
%Y A304157 Cf. A017461, A224454, A224455, A304158.
%K A304157 nonn,easy
%O A304157 1,1
%A A304157 _Emeric Deutsch_, May 07 2018