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

%I #19 Nov 15 2024 14:57:43

%S 12,102,273,525,858,1272,1767,2343,3000,3738,4557,5457,6438,7500,8643,

%T 9867,11172,12558,14025,15573,17202,18912,20703,22575,24528,26562,

%U 28677,30873,33150,35508,37947,40467,43068,45750,48513,51357,54282,57288,60375,63543,66792

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

%C 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 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(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 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 8*a(n) + 25 is a square. - _Bruno Berselli_, May 14 2018

%H Colin Barker, <a href="/A304504/b304504.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 Journal of Mathematical 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>, Journal 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.: 3*(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.: 3*exp(x)*(8 + 60*x + 27*x^2)/2.

%F a(n) = A017197(n)*A017257(n)/2. (End)

%p seq((1/2)*(3*(9*n+8))*(3*n+1), n = 0 .. 40);

%o (PARI) Vec(3*(4 + 22*x + x^2) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 14 2018

%Y Cf. A017197, A017257, A304503.

%K nonn,easy

%O 0,1

%A _Emeric Deutsch_, May 13 2018