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A307307 a(n) is the forgotten index of the Lucas cube Lambda(n).

%I A307307 #12 Apr 02 2019 05:36:47
%S A307307 0,10,30,112,300,840,2044,4864,10944,23960,50908,105840,215748,432656,
%T A307307 855240,1669568,3223404,6162552,11678540,21957440,40988976,76019944,
%U A307307 140155100,256995936,468887700,851538064,1539858168,2773522192,4977094956,8900629800
%N A307307 a(n) is the forgotten index of the Lucas cube Lambda(n).
%C A307307 The Lucas cube Lambda(n) can be defined as the graph whose vertices are the binary strings of length n without either two consecutive 1's or a 1 in the first and in the last position, and in which two vertices are adjacent when their Hamming distance is exactly 1.
%C A307307 The forgotten topological index of a simple connected graph is the sum of the cubes of its vertex degrees.
%C A307307 In the Maple program T(n,k) gives the number of vertices of degree k in the Lucas cube Lambda(n).
%H A307307 B. Furtula and I. Gutman, <a href="https://doi.org/10.1007/s10910-015-0480-z">A forgotten topological index</a>, J. Math. Chem. 53 (4), 1184-1190, 2015.
%H A307307 S. Klavžar, M. Mollard and M. Petkovšek, <a href="https://doi.org/10.1016/j.disc.2011.03.019">The degree sequence of Fibonacci and Lucas cubes</a>, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.
%F A307307 Conjectures from _Colin Barker_, Apr 02 2019: (Start)
%F A307307 G.f.: 2*x^2*(5 - 5*x + 6*x^2 - 4*x^3 + 27*x^4 - 25*x^5 - 6*x^6 + 9*x^8 + 3*x^9) / (1 - x - x^2)^4.
%F A307307 a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 5*a(n-4) + 8*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8) for n>11.
%F A307307 (End)
%e A307307 a(2) = 10 because the Lucas cube Lambda(2) is the path tree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently, the forgotten index is 1^3 + 1^3 + 2^3 = 10.
%p A307307 G:=(1+(1-t)*z + t^2*z^2 + (1-t)*t*z^3 - t*(1-t)^2*z^4)/((1-t*z)*(1-t*z^2)-t*z^3): M:=expand(series(G,z=0,40)): T:=(n,k)->coeff(coeff(M,z,n),t,k): FI:=n->add(T(n,k)*k^3, k=0..n): seq(FI(n), n=1..30);
%Y A307307 Cf. A307181, A307208, A307212.
%K A307307 nonn
%O A307307 1,2
%A A307307 _Emeric Deutsch_, Apr 02 2019