A246175 The hyper-Wiener index of the Fibonacci cube Gamma(n) (n>=1).

1, 5, 23, 89, 325, 1123, 3750, 12174, 38682, 120750, 371478, 1128810, 3394159, 10112987, 29892425, 87737471, 255912115, 742272853, 2142128604, 6153811500, 17605105380, 50174676300, 142501128540, 403422149220, 1138714934125, 3205372562369, 8999834877995, 25209180070037

Offset: 1

Author

Emeric Deutsch, Aug 18 2014

Keywords

• nonn
• easy

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.

Links

• G. G. Cash, Relationship between the Hosoya polynomial and the hyper-Wiener index, Appl. Math. Letters, 15, 2002, 893-895.
• S. Klavzar, M. Mollard, Wiener index and Hosoya polynomial of Fibonacci and Lucas cubes, MATCH Commun. Math. Comput. Chem., 68, 2012, 311-324.
• Index entries for linear recurrences with constant coefficients, signature (6,-6,-19,24,24,-19,-6,6,-1).

Crossrefs

Cf. A246176

Programs

Maple

G := z*(1-z-z^2)/((1+z)^3*(1-3*z+z^2)^3): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, j), j = 1 .. 35);

Mathematica

CoefficientList[Series[z (1-z-z^2)/((1+z)^3(1-3z+z^2)^3),{z,0,30}],z] (* Harvey P. Dale, Mar 05 2019 *)

Formula

G.f.: z*(1-z-z^2)/((1+z)^3*(1-3*z+z^2)^3).

625*a(n) = -1/2*(-1)^n*(74+45*n+5*n^2) -5*(2*A001871(n)-3*A001871(n-1)) +17*A001906(n)-53*A001906(n+1) +50*(2*A246178(n)-3*A246178(n-1)). - R. J. Mathar, Jul 22 2022