A306967 a(n) is the first Zagreb index of the Fibonacci cube Gamma(n).

2, 6, 22, 54, 132, 292, 626, 1290, 2594, 5102, 9864, 18792, 35362, 65838, 121454, 222246, 403788, 728972, 1308562, 2336946, 4154170, 7353310, 12965904, 22781520, 39897410, 69662166, 121292998, 210642966, 364928532, 630794356

Offset: 1

Author

Emeric Deutsch, Mar 26 2019

Keywords

• nonn

Comments

The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and in which two vertices are adjacent when their Hamming distance is exactly 1.

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.

In the Maple program, T(n,k) gives the number of vertices of degree k in the Fibonacci cube Gamma(n) (see A245825).

Examples

			a(2) = 6 because the Fibonacci cube Gamma(2) is the path-tree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently, the Zagreb index is 1^2 + 1^2 + 2^2 = 6 (or (1 + 2) + (2 + 1) = 6).
		

Links

• S. Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
• 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. A245825.

Programs

Maple

T:=(n,k) -> sum(binomial(n - 2*i, k - i) * binomial(i + 1, n - k - i + 1), i = 0..k): seq(add(T(n, k)*k^2, k=1..n), n=1..30);

Formula

a(n) = Sum_{k=1..n} T(n,k)*k^2, where T(n,k) = Sum_{i=0..k} binomial(n-2*i, k-i) * binomial(i+1, n-k-i+1).

Conjectures from Colin Barker, Mar 28 2019: (Start)

G.f.: 2*x*(1 + 2*x^2 - x^3) / (1 - x - x^2)^3.

a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6) for n>6.

(End)