%I #20 Nov 15 2024 18:23:01
%S 20,170,455,875,1430,2120,2945,3905,5000,6230,7595,9095,10730,12500,
%T 14405,16445,18620,20930,23375,25955,28670,31520,34505,37625,40880,
%U 44270,47795,51455,55250,59180,63245,67445,71780,76250,80855,85595,90470,95480,100625,105905,111320
%N a(n) = 5*(3*n+1)*(9*n+8)/2 (n>=0).
%C The second Zagreb index of the single-defect 5-gonal nanocone CNC(5,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(5,n) is M(CNC(5,n); x,y) = 5*x^2*y^2 + 10*n*x^2*y^3 + 5*n*(3*n+1)*x^3*y^3/2.
%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.
%H Colin Barker, <a href="/A304508/b304508.txt">Table of n, a(n) for n = 0..1000</a>
%H Emeric 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, Vol. 6, No. 2, 2015, pp. 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, Vol. 1, No. 1, 2011, pp. 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, Vol. 4, No. 11, 2010, pp. 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.: 5*(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. (End)
%F From _Elmo R. Oliveira_, Nov 15 2024: (Start)
%F E.g.f.: 5*exp(x)*(8 + 60*x + 27*x^2)/2.
%F a(n) = 5*A016777(n)*A017257(n)/2. (End)
%p seq((1/2)*(5*(3*n+1))*(9*n+8), n = 0 .. 40);
%t Array[5 (3 # + 1) (9 # + 8)/2 &, 41, 0] (* or *)
%t LinearRecurrence[{3, -3, 1}, {20, 170, 455}, 41] (* or *)
%t CoefficientList[Series[5 (4 + 22 x + x^2)/(1 - x)^3, {x, 0, 40}], x] (* _Michael De Vlieger_, May 14 2018 *)
%o (PARI) a(n) = 5*(3*n+1)*(9*n+8)/2; \\ _Altug Alkan_, May 14 2018
%o (PARI) Vec(5*(4 + 22*x + x^2) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 14 2018
%Y Cf. A016777, A017257, A304507.
%K nonn,easy,changed
%O 0,1
%A _Emeric Deutsch_, May 14 2018