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%I #17 Jan 29 2023 11:27:10
%S 324,1404,3240,5832,9180,13284,18144,23760,30132,37260,45144,53784,
%T 63180,73332,84240,95904,108324,121500,135432,150120,165564,181764,
%U 198720,216432,234900,254124,274104,294840,316332,338580,361584,385344,409860,435132,461160,487944,515484,543780,572832,602640,633204
%N a(n) = 378*n^2 - 54*n (n>=1).
%C a(n) is the first Zagreb index of the silicate network SL(n), defined pictorially in the Javaid et al. reference (Fig. 1, where SL(2) is shown) or in Liu et al. reference (Fig. 3, where again SL(2) is shown).
%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 SL(n) is M(SL(n);x,y) = 6*n*x^3*y^3 + (18*n^2 + 6*n)*x^3*y^6 + (18*n^2 - 12*n)*x^6*y^6 (n>=2).
%C 14*a(n)/27 + 1; is a square. - _Muniru A Asiru_, May 27 2018
%H Muniru A Asiru, <a href="/A305070/b305070.txt">Table of n, a(n) for n = 1..5000</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 J. Math. Chemistry, 6, No. 2, 2015, 93-102.
%H M. Javaid and C. Y. Jung, <a href="https://doi.org/10.12732/ijpam.v115i1.11">M-polynomials and topological indices of silicate and oxide networks</a>, International J. Pure and Applied Math., 115, No. 1, 2017, 129-152.
%H J.-B. Liu, S. Wang, C. Wang, and S. Hayat, <a href="https://doi.org/10.1049/iet-cta.2016.1237">Further results on computation of topological indices of certain networks</a>, IET Control Theory Appl., 11, No. 13, 2017, 2065-2071.
%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 26 2018: (Start)
%F G.f.: 108*x*(3 + 4*x) / (1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
%F (End)
%p seq(378*n^2 - 54*n, n = 1 .. 50);
%t Table[378n^2-54n,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,324,1404},50] (* _Harvey P. Dale_, Jan 29 2023 *)
%o (PARI) Vec(108*x*(3 + 4*x) / (1 - x)^3 + O(x^50)) \\ _Colin Barker_, May 26 2018
%o (GAP) List([1..50], n->378*n^2-54*n); # _Muniru A Asiru_, May 27 2018
%Y Cf. A305071.
%K nonn,easy
%O 1,1
%A _Emeric Deutsch_, May 25 2018