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%I #32 Apr 16 2023 08:36:48
%S 162,1854,5490,11070,18594,28062,39474,52830,68130,85374,104562,
%T 125694,148770,173790,200754,229662,260514,293310,328050,364734,
%U 403362,443934,486450,530910,577314,625662,675954,728190,782370,838494,896562,956574,1018530,1082430,1148274,1216062,1285794,1357470,1431090,1506654
%N a(n) = 972*n^2 - 1224*n + 414 with n > 0.
%C a(n) provides the second Zagreb index of the HcDN1(n) network (see Fig. 3 in the Hayat et al. paper).
%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 HcDN1(n) is M(HcDN1(n); x,y) = 6x^3*y^3 + 12(n-1)x^3*y^5 + 6nx^3*y^6 + 18(n-1)x^5*y^6 + (27n^2 - 57n + 30)x^6*y^6. - _Emeric Deutsch_, May 11 2018
%H Colin Barker, <a href="/A304166/b304166.txt">Table of n, a(n) for n = 1..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, 6, No. 2, 2015, 93-102.
%H S. Hayat, M. A. Malik, and M. Imran, <a href="http://www.romjist.ro/content/pdf/03-mimran.pdf">Computing topological indices of honeycomb derived networks</a>, Romanian J. of Information Science and Technology, 18, No. 2, 2015, 144-165.
%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 10 2018: (Start)
%F G.f.: 18*x*(9 + 76*x + 23*x^2)/(1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
%F E.g.f.: 18*(exp(x)*(23 - 14*x + 54*x^2) - 23). - _Stefano Spezia_, Apr 15 2023
%p seq(972*n^2-1224*n+414, n = 1 .. 40);
%o (PARI) a(n) = 972*n^2-1224*n+414; \\ _Altug Alkan_, May 09 2018
%o (PARI) Vec(18*x*(9 + 76*x + 23*x^2) / (1 - x)^3 + O(x^60)) \\ _Colin Barker_, May 10 2018
%Y Cf. A304163, A304164, A304165.
%K nonn,easy
%O 1,1
%A _Emeric Deutsch_, May 09 2018
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