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A304505 a(n) = 4*(n+1)*(9*n+4).

%I A304505 #31 Nov 15 2024 18:08:41
%S A304505 16,104,264,496,800,1176,1624,2144,2736,3400,4136,4944,5824,6776,7800,
%T A304505 8896,10064,11304,12616,14000,15456,16984,18584,20256,22000,23816,
%U A304505 25704,27664,29696,31800,33976,36224,38544,40936,43400,45936,48544,51224,53976,56800,59696
%N A304505 a(n) = 4*(n+1)*(9*n+4).
%C A304505 a(n) is the first Zagreb index of the single-defect 4-gonal nanocone CNC(4,n) (see definition in the Doslic et al. reference, p. 27).
%C A304505 The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
%C A304505 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 A304505 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 A304505 9*a(n) + 25 is a square. - _Bruno Berselli_, May 14 2018
%H A304505 Colin Barker, <a href="/A304505/b304505.txt">Table of n, a(n) for n = 0..1000</a>
%H A304505 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 A304505 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 A304505 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 A304505 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A304505 From _Colin Barker_, May 14 2018: (Start)
%F A304505 G.f.: 8*(2 + 7*x)/(1 - x)^3.
%F A304505 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%F A304505 From _Elmo R. Oliveira_, Nov 15 2024: (Start)
%F A304505 E.g.f.: 4*exp(x)*(4 + 22*x + 9*x^2).
%F A304505 a(n) = 8*A062708(n+1) = A017209(n)*A008586(n+1). (End)
%p A304505 seq((4*(n+1))*(9*n+4), n = 0 .. 40);
%o A304505 (PARI) a(n) = 4*(n+1)*(9*n+4); \\ _Altug Alkan_, May 14 2018
%o A304505 (GAP) List([0..50],n->4*(n+1)*(9*n+4)); # _Muniru A Asiru_, May 14 2018
%o A304505 (PARI) Vec(8*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 14 2018
%Y A304505 Cf. A008586, A017209, A062708, A304506.
%K A304505 nonn,easy
%O A304505 0,1
%A A304505 _Emeric Deutsch_, May 14 2018