A246173 Triangle read by rows: T(n,k) is the number of vertex pairs at distance k of the Fibonacci cube Gamma(n) (1<=k<=n).

1, 2, 1, 5, 4, 1, 10, 11, 6, 1, 20, 28, 21, 8, 1, 38, 64, 62, 35, 10, 1, 71, 140, 164, 120, 53, 12, 1, 130, 293, 402, 360, 210, 75, 14, 1, 235, 596, 935, 984, 708, 340, 101, 16, 1, 420, 1183, 2086, 2517, 2142, 1280, 518, 131, 18, 1, 744, 2304, 4507, 6120, 5991, 4260, 2164, 752, 165, 20, 1

Offset: 1

Author

Emeric Deutsch, Aug 18 2014

Keywords

• nonn
• tabl

Comments

The Fibonacci cube Gamma(n) is obtained from the n-cube Q(n) by removing all the vertices that contain two consecutive 1s.

The entries in row n are the coefficients of the Hosoya polynomial of the Fibonacci cube Gamma(n).

T(n,1) = A001629(n+1) = number of edges in Gamma(n).

Sum of entries in row n = A191797(n+2).

Sum(k*T(n,k), k>=1) = A238419(n) = the Wiener index of Gamma(n).

Examples

			Row 2 is 2,1. Indeed, Gamma(2) is the path-tree P(3) having vertex-pair distances 1,1, and 2.
Triangle starts:
1;
2,1;
5,4,1;
10,11,6,1;
20,28,21,8,1;
		

Links

• S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
• S. Klavzar, M. Mollard, Wiener index and Hosoya polynomial of Fibonacci and Lucas cubes, MATCH Commun. Math. Comput. Chem., 68, 2012, 311-324.
• Eric Weisstein's World of Mathematics, Fibonacci Cube Graph

Crossrefs

Cf. A001629, A191797, A238419, A246174

Programs

Maple

g := t*z/((1-z-z^2-t*z-t*z^2+t*z^3)*(1-z-z^2)): gserz := simplify(series(g, z = 0, 20)): for j to 18 do H[j] := sort(coeff(gserz, z, j)) end do: for j to 13 do seq(coeff(H[j], t, k), k = 1 .. j) end do; # yields sequence in triangular form

Mathematica

Rest /@ Rest[CoefficientList[CoefficientList[Series[t z/((1 - z - z^2 - t z - t z^2 + t z^3) (1 - z - z^2)), {z, 0, 10}, {t, 0, 5}], z], t]] // Flatten (* Eric W. Weisstein, Dec 11 2017 *)
DeleteCases[CoefficientList[Series[t z/((1 - z - z^2 - t z - t z^2 + t z^3) (1 - z - z^2)), {z, 0, 10}], {z, t}], 0, {2}] // Flatten (* Eric W. Weisstein, Dec 11 2017 *)

Formula

G.f.: tz/((1-z-z^2-tz-tz^2+tz^3)(1-z-z^2)). Derived from Theorem 4.1 of the Klavzar-Mollard reference in which the g.f. of the ordered Hosoya polynomials is given.