The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k ≥ 0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonac...

Spherical harmonic cross-correlation is a robust registration technique that uses the normals of two overlapping point clouds to bring them into coarse rotational alignment. This registration technique however has a high computational cost as spherical harmonics need to be calculated for every normal. By binning the normals, the computational efficiency is improved as the spherical harmonics ca...

In this project, we study a randomized variant of Fibonacci heaps where instead of using mark bits, one flips coins in order to determine whether to cascade bringing nodes into the root list. Although it seems intuitive that such heaps should have the same expected performance as standard Fibonacci heaps—and Karger has conjectured as such—the only previous work was an O(log s) upper bound using...

A non-abelian finite group is called sequenceable if for some positive integer , is -generated ( ) and there exist integers such that every element of is a term of the -step generalized Fibonacci sequence , , , . A remarkable application of this definition may be find on the study of random covers in the cryptography. The 2-step generalized sequences for the dihedral groups studi...

Let Γn and Λn be the n-dimensional Fibonacci cube and Lucas cube, respectively. The domination number γ of Fibonacci cubes and Lucas cubes is studied. In particular it is proved that γ(Λn) is bounded below by ⌈ Ln−2n n−3 ⌉ , where Ln is the n-th Lucas number. The 2-packing number ρ of these cubes is also studied. It is proved that ρ(Γn) is bounded below by 2 blg nc 2 −1 and the exact values of ...

The explicite formulas for Möbius function and some other important elements of the incidence algebra are delivered. For that to do one uses Kwaśniewski’s construction of his Fibonacci cobweb poset in the plane grid coordinate system. 1 Fibonacci cobweb poset The Fibonacci cobweb poset P has been invented by A.K.Kwaśniewski in [1, 2, 3] for the purpose of finding combinatorial interpretation of...

In this paper, we consider the generalized Fibonacci and Pell Sequences and then show the relationships between the generalized Fibonacci and Pell sequences, and the Hessenberg permanents and determinants. 1. Introduction The Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 1 Fn+1 = Fn + Fn 1 (1.1) where F0 = 0; F1 = 1: The Pell Sequence, fPng ; is de…ned by the recurrence...

This work aims at showing the relevance and the applications possibilities of the Fibonacci sequence, and also its q-deformed or “quantum” extension, in the study of the genetic code(s). First, after the presentation of a new formula, an indexed double Fibonacci sequence, comprising the first six Fibonacci numbers, is shown to describe the 20 amino acids multiplets and their degeneracy as well ...

We consider a q-analogue of the cube polynomial of Fibonacci cubes. These bivariate polynomials satisfy a recurrence relation similar to the standard one. They refine the count of the number of hypercubes of a given dimension in Fibonacci cubes by keeping track of the distances of the hypercubes to the all 0 vertex. For q = 1, they specialize to the standard cube polynomials. We also investigat...

A prime p is called a Fibonacci-Wieferich prime if Fp−( p5 ) ≡ 0 (mod p2), where Fn is the nth Fibonacci number. We report that there exist no such primes p < 2 × 1014. A prime p is called a Wolstenholme prime if (2p−1 p−1 ) ≡ 1 (mod p4). To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes p < 109. Wolstenholme, in 1862, proved...