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%I #37 Apr 15 2023 23:32:30
%S 90,726,2010,3942,6522,9750,13626,18150,23322,29142,35610,42726,50490,
%T 58902,67962,77670,88026,99030,110682,122982,135930,149526,163770,
%U 178662,194202,210390,227226,244710,262842,281622,301050,321126,341850,363222,385242,407910,431226,455190,479802,505062
%N a(n) = 324*n^2 - 336*n + 102 (n >= 1).
%C a(n) is the first Zagreb index of the HcDN1(n) network (see Fig. 3 in the Hayat et al. manuscript).
%C 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 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="/A304165/b304165.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.: 6*x*(15 + 76*x + 17*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.: 6*(exp(x)*(17 - 2*x + 54*x^2) - 17). - _Stefano Spezia_, Apr 15 2023
%p seq(324*n^2-336*n+102,n=1..40);
%t Table[324n^2-336n+102,{n,40}] (* or *) LinearRecurrence[{3,-3,1},{90,726,2010},40] (* _Harvey P. Dale_, Apr 12 2020 *)
%o (PARI) a(n) = 324*n^2-336*n+102; \\ _Altug Alkan_, May 09 2018
%o (PARI) Vec(6*x*(15 + 76*x + 17*x^2) / (1 - x)^3 + O(x^60)) \\ _Colin Barker_, May 10 2018
%o (GAP) List([1..40],n->324*n^2-336*n+102); # _Muniru A Asiru_, May 10 2018
%Y Cf. A304163, A304164, A304166.
%K nonn,easy
%O 1,1
%A _Emeric Deutsch_, May 09 2018
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