A307208 a(n) is the forgotten index of the Fibonacci cube Gamma(n).

2, 10, 52, 158, 466, 1192, 2914, 6722, 14972, 32286, 67914, 139824, 282754, 562970, 1105892, 2146846, 4124258, 7849496, 14815202, 27752338, 51632620, 95465502, 175508250, 320981472, 584214530, 1058602666, 1910305300, 3434059166, 6151218034, 10981579528

Offset: 1

Author

Emeric Deutsch, Mar 28 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 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 Fibonacci cube Gamma(n) (see A245825).

Examples

			a(2) = 10 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 forgotten index is 1^3 + 1^3 + 2^3 = 10.
		

Links

• B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (4), 1184-1190, 2015.
• 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, A306967, A307157.

Programs

Maple

T := (n,k) -> add(binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1), i=0..k):
seq(add(T(n,k)*k^3, k=1..n), n=1..30);

PARI

T(n,k) = sum(i=0, k, binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1));
a(n) = sum(k=1, n, T(n,k)*k^3); \\ Michel Marcus, Mar 30 2019

Formula

a(n) = Sum_{k=1..n} T(n,k)*k^3 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 29 2019: (Start)

G.f.: 2*x*(1 + x + 8*x^2 - 7*x^3 + 4*x^4 - 3*x^5 + 3*x^6) / (1 - x - x^2)^4.

a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 5*a(n-4) + 8*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8) for n>8.

(End)