cp's OEIS Frontend

A307181 a(n) is the first Zagreb index of the Lucas cube Lambda(n).

%I A307181 #17 Apr 02 2019 08:46:35
%S A307181 0,6,12,40,90,216,462,976,1980,3940,7678,14736,27898,52220,96780,
%T A307181 177824,324258,587304,1057350,1893320,3373692,5984924,10574342,
%U A307181 18613920,32654450,57106036,99576972,173166616,300385770,519849720
%N A307181 a(n) is the first Zagreb index of the Lucas cube Lambda(n).
%C A307181 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 A307181 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 A307181 In the Maple program, T(n,k) gives the number of vertices of degree k in the Lucas cube Lambda(n).
%H A307181 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 A307181 Conjectures from _Colin Barker_, Mar 28 2019: (Start)
%F A307181 G.f.: 2*x^2*(3 - 3*x + 2*x^2 + 3*x^4 - 2*x^5 - x^6) / (1 - x - x^2)^3.
%F A307181 a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6) for n>8.
%F A307181 (End)
%e A307181 a(2) = 6 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 first Zagreb index is 1^2 + 1^2 + 2^2 = 6 (or (1 + 2) + (2 + 1) = 6).
%p A307181 L:=(1+(1-y)*x + x^2*y^2 + (1-y)*x^3*y-(1-y)^2*x^4*y)/((1-x*y)*(1-x^2*y)-x^3*y): M:=expand(series(L,x=0,40)): T:=(n,k)->coeff(coeff(M,x,n),y,k): Z1:=n->add(T(n,k)*k^2, k=0..n): seq(Z1(n), n=1..35);
%Y A307181 Cf. A306967, A307307, A307212
%K A307181 nonn
%O A307181 1,2
%A A307181 _Emeric Deutsch_, Mar 28 2019