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%I #51 Sep 08 2022 08:46:17
%S 0,22,52,90,136,190,252,322,400,486,580,682,792,910,1036,1170,1312,
%T 1462,1620,1786,1960,2142,2332,2530,2736,2950,3172,3402,3640,3886,
%U 4140,4402,4672,4950,5236,5530,5832,6142,6460,6786,7120,7462,7812,8170,8536,8910,9292,9682
%N a(n) = 4*n^2 + 18*n.
%C For n>=3, a(n) is the first Zagreb index of the double-wheel graph DW[n]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
%C The double-wheel graph DW[n] consists of two cycles C[n], whose vertices are connected to an additional vertex.
%C The M-polynomial of the double-wheel graph DW[n] is M(DW[n],x,y)= 2*n*x^3*y^3 + 2*n*x^3*y^{2*n}.
%C 4*a(n) + 81 is a square. - _Bruno Berselli_, May 08 2018
%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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F O.g.f.: 2*x*(11 - 7*x)/(1 - x)^3.
%F E.g.f.: 2*x*(11 + 2*x)*exp(x).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F a(n) = 2*A139576(n).
%e a(3) = 90. Indeed, the double-wheel graph DW[3] has 6 edges with end-point degrees 3,3 and 6 edges with end-point degrees 3,6. Then the first Zagreb index is 6*6 + 6*9 = 90.
%p seq(4*n^2+18*n,n = 0..50);
%t Table[4 n^2 + 18 n, {n, 0, 50}] (* _Vincenzo Librandi_, Nov 09 2016 *)
%t LinearRecurrence[{3,-3,1},{0,22,52},50] (* _Harvey P. Dale_, Mar 01 2022 *)
%o (Magma) [4*n^2+18*n: n in [0..50]]; // _Vincenzo Librandi_, Nov 09 2016
%o (PARI) a(n)=4*n^2+18*n \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A139576, A277980.
%Y Subsequence of A028569.
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Nov 08 2016
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