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A306967 a(n) is the first Zagreb index of the Fibonacci cube Gamma(n). 2
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..30.

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

Cf. A245825.

Sequence in context: A147800 A246624 A212872 * A324913 A027561 A321626

Adjacent sequences:  A306964 A306965 A306966 * A306968 A306969 A306970

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Mar 26 2019

STATUS

approved

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Last modified April 3 14:34 EDT 2020. Contains 333197 sequences. (Running on oeis4.)