OFFSET
1,1
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).
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.
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)
EXAMPLE
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).
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);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 26 2019
STATUS
approved