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%I #21 Aug 15 2022 13:51:53
%S 16,136,364,700,1144,1696,2356,3124,4000,4984,6076,7276,8584,10000,
%T 11524,13156,14896,16744,18700,20764,22936,25216,27604,30100,32704,
%U 35416,38236,41164,44200,47344,50596,53956,57424,61000,64684,68476,72376,76384,80500,84724,89056
%N a(n) = 2*(3*n+1)*(9*n+8).
%C 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).
%C 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 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.
%C 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.
%C 6*a(n) + 25 is a square. - _Bruno Berselli_, May 14 2018
%H Colin Barker, <a href="/A304506/b304506.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 T. Doslic and M. Saheli, <a href="http://dx.doi.org/10.22061/jmns.2011.460">Augmented eccentric connectivity index of single-defect nanocones</a>, J. of Mathematical Nanoscience, 1, No. 1, 2011, 25-31.
%H A. Khaksar, M. Ghorbani, and H. R. Maimani, <a href="https://oam-rc.inoe.ro/download.php?idu=1353">On atom bond connectivity and GA indices of nanocones</a>, Optoelectronics and Advanced Materials - Rapid Communications, 4, No. 11, 2010, 1868-1870.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F From _Colin Barker_, May 14 2018: (Start)
%F G.f.: 4*(4 + 22*x + x^2) / (1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
%F (End)
%p seq((2*(9*n+8))*(3*n+1), n = 0 .. 40);
%t Table[2(3n+1)(9n+8),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{16,136,364},50] (* _Harvey P. Dale_, Aug 15 2022 *)
%o (PARI) a(n) = 2*(3*n+1)*(9*n+8); \\ _Altug Alkan_, May 14 2018
%o (GAP) List([0..50],n->2*(3*n+1)*(9*n+8)); # _Muniru A Asiru_, May 14 2018
%o (PARI) Vec(4*(4 + 22*x + x^2) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 14 2018
%Y Cf. A304505.
%K nonn,easy
%O 0,1
%A _Emeric Deutsch_, May 14 2018
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