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A304504 a(n) = 3*(3*n+1)*(9*n+8)/2.

%I A304504 #19 Nov 15 2024 14:57:43
%S A304504 12,102,273,525,858,1272,1767,2343,3000,3738,4557,5457,6438,7500,8643,
%T A304504 9867,11172,12558,14025,15573,17202,18912,20703,22575,24528,26562,
%U A304504 28677,30873,33150,35508,37947,40467,43068,45750,48513,51357,54282,57288,60375,63543,66792
%N A304504 a(n) = 3*(3*n+1)*(9*n+8)/2.
%C A304504 The second Zagreb index of the single-defect 3-gonal nanocone CNC(3,n) (see definition in the Doslic et al. reference, p. 27).
%C A304504 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 A304504 The M-polynomial of CNC(3,n) is M(CNC(3,n); x,y) = 3*x^2*y^2 + 6*n*x^2*y^3 + 3*n*(3*n+1)*x^3*y^3/2.
%C A304504 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 A304504 8*a(n) + 25 is a square. - _Bruno Berselli_, May 14 2018
%H A304504 Colin Barker, <a href="/A304504/b304504.txt">Table of n, a(n) for n = 0..1000</a>
%H A304504 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 Journal of Mathematical Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
%H A304504 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>, Journal of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.
%H A304504 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, Vol. 4, No. 11, 2010, pp. 1868-1870.
%H A304504 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A304504 From _Colin Barker_, May 14 2018: (Start)
%F A304504 G.f.: 3*(4 + 22*x + x^2)/(1 - x)^3.
%F A304504 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%F A304504 From _Elmo R. Oliveira_, Nov 15 2024: (Start)
%F A304504 E.g.f.: 3*exp(x)*(8 + 60*x + 27*x^2)/2.
%F A304504 a(n) = A017197(n)*A017257(n)/2. (End)
%p A304504 seq((1/2)*(3*(9*n+8))*(3*n+1), n = 0 .. 40);
%o A304504 (PARI) Vec(3*(4 + 22*x + x^2) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 14 2018
%Y A304504 Cf. A017197, A017257, A304503.
%K A304504 nonn,easy
%O A304504 0,1
%A A304504 _Emeric Deutsch_, May 13 2018