A307212 -id:A307212 - cp's OEIS frontend

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

Emeric Deutsch, Mar 28 2019

nonn

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).

			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).

S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Cf. A306967, A307307, A307212

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);

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)

A307307 a(n) is the forgotten index of the Lucas cube Lambda(n).

Original entry on oeis.org

0, 10, 30, 112, 300, 840, 2044, 4864, 10944, 23960, 50908, 105840, 215748, 432656, 855240, 1669568, 3223404, 6162552, 11678540, 21957440, 40988976, 76019944, 140155100, 256995936, 468887700, 851538064, 1539858168, 2773522192, 4977094956, 8900629800

Offset: 1

Emeric Deutsch, Apr 02 2019

nonn

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 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 Lucas cube Lambda(n).

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

B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (4), 1184-1190, 2015.
S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

Cf. A307181, A307208, A307212.

G:=(1+(1-t)*z + t^2*z^2 + (1-t)*t*z^3 - t*(1-t)^2*z^4)/((1-t*z)*(1-t*z^2)-t*z^3): M:=expand(series(G,z=0,40)): T:=(n,k)->coeff(coeff(M,z,n),t,k): FI:=n->add(T(n,k)*k^3, k=0..n): seq(FI(n), n=1..30);

Conjectures from Colin Barker, Apr 02 2019: (Start)
G.f.: 2*x^2*(5 - 5*x + 6*x^2 - 4*x^3 + 27*x^4 - 25*x^5 - 6*x^6 + 9*x^8 + 3*x^9) / (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>11.
(End)

cp's OEIS Frontend

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

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