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A304158 a(n) is the second Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference (Fig. 3).

%I A304158 #28 Jan 05 2025 19:51:41
%S A304158 24,84,144,204,264,324,384,444,504,564,624,684,744,804,864,924,984,
%T A304158 1044,1104,1164,1224,1284,1344,1404,1464,1524,1584,1644,1704,1764,
%U A304158 1824,1884,1944,2004,2064,2124,2184,2244,2304,2364,2424,2484,2544,2604,2664,2724,2784,2844,2904,2964
%N A304158 a(n) is the second Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference (Fig. 3).
%C A304158 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.
%C A304158 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.
%H A304158 Colin Barker, <a href="/A304158/b304158.txt">Table of n, a(n) for n = 1..1000</a>
%H A304158 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 A304158 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 A304158 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 A304158 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 A304158 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A304158 a(n) = 60*n - 36.
%F A304158 a(n) = 12 * A016873(n-1). - _Alois P. Heinz_, May 09 2018
%F A304158 From _Bruno Berselli_, May 09 2018: (Start)
%F A304158 O.g.f.: 12*x*(2 + 3*x)/(1 - x)^2.
%F A304158 E.g.f.: 12*(3 - 3*exp(x) + 5*x*exp(x)).
%F A304158 a(n) = 2*a(n-1) - a(n-2).
%F A304158 a(n) = A008594(5*n-3) = A017317(6*n-4) = A072710(4*n-2) = A139245(3*n-1). (End)
%e A304158 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.
%p A304158 seq(60*n - 36, n = 1 .. 40);
%o A304158 (Julia) [60*n-36 for n in 1:50] |> println # _Bruno Berselli_, May 09 2018
%o A304158 (PARI) a(n) = 60*n-36; \\ _Altug Alkan_, May 09 2018
%o A304158 (PARI) Vec(12*x*(2 + 3*x)/(1 - x)^2 + O(x^40)) \\ _Colin Barker_, May 23 2018
%Y A304158 Cf. A016873, A304157.
%Y A304158 Subsequence of A121024.
%K A304158 nonn,easy
%O A304158 1,1
%A A304158 _Emeric Deutsch_, May 08 2018