A307212 a(n) is the Narumi-Katayama index of the Lucas cube Lambda(n).

0, 2, 3, 256, 38880, 1289945088, 42855402240000000, 605828739547255327948800000000, 13263549731442762279026688000000000000000000000000000, 1334793240853871268746431553848403294648071618560000000000000000000000000000000000000000000

Offset: 1

Author

Emeric Deutsch, Mar 28 2019

Keywords

• nonn

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 Narumi-Katayama index of a connected graph is the product of the degrees of the vertices of the graph.

Examples

			a(2) = 2 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 Narumi-Katayama index is 1*1*2 = 2.
		

Links

• I. Gutman and M. Ghorbani, Some properties of the Narumi-Katayama index, Applied Mathematics Letters, Vol. 25, No. 10 (2012), 1435-1438.
• S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Crossrefs

Cf. A307157, A307181, A307307.

Programs

Maple

G := (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):
g := expand(series(G, x=0, 40)): T := (n, k) -> coeff(coeff(g, x, n), y, k):
a := n -> mul(k^T(n, k), k=0..n): lprint(seq(a(n), n=1..10));