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 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 (list; graph; refs; listen; history; text; internal format)
 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 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 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) = 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) EXAMPLE 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) 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 Cf. A017461, A224454, A224455, A304158. Sequence in context: A043934 A087406 A319972 * A051876 A069174 A124717 Adjacent sequences: A304154 A304155 A304156 * A304158 A304159 A304160 KEYWORD nonn,easy AUTHOR Emeric Deutsch, May 07 2018 STATUS approved

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Last modified November 28 15:31 EST 2023. Contains 367419 sequences. (Running on oeis4.)