

A307307


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


2



0, 10, 30, 112, 300, 840, 2044, 4864, 10944, 23960, 50908, 105840, 215748, 432656, 855240, 1669568, 3223404, 6162552, 11678540, 21957440, 40988976, 76019944, 140155100, 256995936, 468887700, 851538064, 1539858168, 2773522192, 4977094956, 8900629800
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OFFSET

1,2


COMMENTS

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.
The forgotten topological index of a simple connected graph is the sum of the cubes of its vertex degrees.
In the Maple program T(n,k) gives the number of vertices of degree k in the Lucas cube Lambda(n).


LINKS



FORMULA

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.
a(n) = 4*a(n1)  2*a(n2)  8*a(n3) + 5*a(n4) + 8*a(n5)  2*a(n6)  4*a(n7)  a(n8) for n>11.
(End)


EXAMPLE

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.


MAPLE

G:=(1+(1t)*z + t^2*z^2 + (1t)*t*z^3  t*(1t)^2*z^4)/((1t*z)*(1t*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);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



