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A304158
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a(n) is the second Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference (Fig. 3).
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2
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24, 84, 144, 204, 264, 324, 384, 444, 504, 564, 624, 684, 744, 804, 864, 924, 984, 1044, 1104, 1164, 1224, 1284, 1344, 1404, 1464, 1524, 1584, 1644, 1704, 1764, 1824, 1884, 1944, 2004, 2064, 2124, 2184, 2244, 2304, 2364, 2424, 2484, 2544, 2604, 2664, 2724, 2784, 2844, 2904, 2964
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OFFSET
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1,1
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COMMENTS
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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.
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.
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LINKS
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FORMULA
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a(n) = 60*n - 36.
O.g.f.: 12*x*(2 + 3*x)/(1 - x)^2.
E.g.f.: 12*(3 - 3*exp(x) + 5*x*exp(x)).
a(n) = 2*a(n-1) - a(n-2).
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EXAMPLE
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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.
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MAPLE
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seq(60*n - 36, n = 1 .. 40);
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PROG
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(Julia) [60*n-36 for n in 1:50] |> println # Bruno Berselli, May 09 2018
(PARI) Vec(12*x*(2 + 3*x)/(1 - x)^2 + O(x^40)) \\ Colin Barker, May 23 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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