cp's OEIS Frontend

A304503 a(n) = 3*(n+1)*(9*n+4).

%I A304503 #28 Nov 16 2024 10:23:07
%S A304503 12,78,198,372,600,882,1218,1608,2052,2550,3102,3708,4368,5082,5850,
%T A304503 6672,7548,8478,9462,10500,11592,12738,13938,15192,16500,17862,19278,
%U A304503 20748,22272,23850,25482,27168,28908,30702,32550,34452,36408,38418,40482,42600,44772
%N A304503 a(n) = 3*(n+1)*(9*n+4).
%C A304503 The first Zagreb index of the single-defect 3-gonal nanocone CNC(3,n) (see definition in the Doslic et al. reference, p. 27).
%C A304503 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 A304503 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 A304503 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 A304503 12*a(n) + 25 is a square. - _Bruno Berselli_, May 14 2018
%H A304503 Colin Barker, <a href="/A304503/b304503.txt">Table of n, a(n) for n = 0..1000</a>
%H A304503 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 A304503 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 A304503 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 A304503 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A304503 From _Colin Barker_, May 14 2018: (Start)
%F A304503 G.f.: 6*(2 + 7*x)/(1 - x)^3.
%F A304503 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%F A304503 From _Elmo R. Oliveira_, Nov 15 2024: (Start)
%F A304503 E.g.f.: 3*exp(x)*(4 + 22*x + 9*x^2).
%F A304503 a(n) = 6*A062708(n+1) = A017209(n)*A008585(n+1). (End)
%p A304503 seq((3*(n+1))*(9*n+4), n = 0 .. 40);
%o A304503 (PARI) Vec(6*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 14 2018
%Y A304503 Cf. A008585, A017209, A062708, A304504.
%K A304503 nonn,easy
%O A304503 0,1
%A A304503 _Emeric Deutsch_, May 13 2018