A307181 a(n) is the first Zagreb index of the Lucas cube Lambda(n).

0, 6, 12, 40, 90, 216, 462, 976, 1980, 3940, 7678, 14736, 27898, 52220, 96780, 177824, 324258, 587304, 1057350, 1893320, 3373692, 5984924, 10574342, 18613920, 32654450, 57106036, 99576972, 173166616, 300385770, 519849720

Offset

1,2

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 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 Lucas cube Lambda(n).

Links

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

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

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

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

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

(End)

Example

a(2) = 6 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 first Zagreb index is 1^2 + 1^2 + 2^2 = 6 (or (1 + 2) + (2 + 1) = 6).

Maple

L:=(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): M:=expand(series(L, x=0, 40)): T:=(n, k)->coeff(coeff(M, x, n), y, k): Z1:=n->add(T(n, k)*k^2, k=0..n): seq(Z1(n), n=1..35);

Crossrefs

Cf. A306967, A307307, A307212

Sequence in context: A215179 A128453 A185616 * A052747 A007121 A356910

Adjacent sequences: A307178 A307179 A307180 * A307182 A307183 A307184

Keyword

nonn

Author

Emeric Deutsch, Mar 28 2019

Status

approved