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 A304506 a(n) = 2*(3*n+1)*(9*n+8). 2
 16, 136, 364, 700, 1144, 1696, 2356, 3124, 4000, 4984, 6076, 7276, 8584, 10000, 11524, 13156, 14896, 16744, 18700, 20764, 22936, 25216, 27604, 30100, 32704, 35416, 38236, 41164, 44200, 47344, 50596, 53956, 57424, 61000, 64684, 68476, 72376, 76384, 80500, 84724, 89056 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS a(n) is the second Zagreb index of the single-defect 4-gonal nanocone CNC(4,n) (see definition in the Doslic et al. reference, p. 27). 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 CNC(4,n) is M(CNC(4,n);x,y) = 4*x^2*y^2 + 8*n*x^2*y^3 + 2*n*(3*n+1)*x^3*y^3. More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n); x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2. 6*a(n) + 25 is a square. - Bruno Berselli, May 14 2018 LINKS Colin Barker, Table of n, a(n) for n = 0..1000 E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102. T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, J. of Mathematical Nanoscience, 1, No. 1, 2011, 25-31. A. Khaksar, M. Ghorbani, and H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials - Rapid Communications, 4, No. 11, 2010, 1868-1870. Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA From Colin Barker, May 14 2018: (Start) G.f.: 4*(4 + 22*x + x^2) / (1 - x)^3. a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End) MAPLE seq((2*(9*n+8))*(3*n+1), n = 0 .. 40); MATHEMATICA Table[2(3n+1)(9n+8), {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {16, 136, 364}, 50] (* Harvey P. Dale, Aug 15 2022 *) PROG (PARI) a(n) = 2*(3*n+1)*(9*n+8); \\ Altug Alkan, May 14 2018 (GAP) List([0..50], n->2*(3*n+1)*(9*n+8)); # Muniru A Asiru, May 14 2018 (PARI) Vec(4*(4 + 22*x + x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018 CROSSREFS Cf. A304505. Sequence in context: A219904 A253303 A139616 * A187175 A302319 A303013 Adjacent sequences:  A304503 A304504 A304505 * A304507 A304508 A304509 KEYWORD nonn,easy AUTHOR Emeric Deutsch, May 14 2018 STATUS approved

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)