This paper describes the evolution of recursive functions within an Object-Oriented Genetic Programming (OOGP) system. We evolved general solutions to factorial, Fibonacci, exponentiation, even-nParity, and nth-3. We report the computational effort required to evolve these methods and provide a comparison between crossover and mutation variation operators, and also undirected random search. We ...

The Fibonacci cube of dimension n, denoted as Γn, is the subgraph of n-cube Qn induced by vertices with no consecutive 1’s. We study the maximum number of disjoint subgraphs in Γn isomorphic to Qk, and denote this number by qk(n). We prove several recursive results for qk(n), in particular we prove that qk(n) = qk−1(n − 2) + qk(n − 3). We also prove a closed formula in which qk(n) is given in t...

We provide a formula for the number of edges of the Hasse diagram of the independent subsets of the hth power of a path ordered by inclusion. For h = 1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself.

In this paper, we consider pattern avoidance in k-ary heaps, where the permutation associated with the heap is found by recording the nodes as they are encountered in a breadth-first search. We enumerate heaps that avoid patterns of length 3 and collections of patterns of length 3, first with binary heaps and then more generally with k-ary heaps.

The Chord protocol is the best known example of implementation of logarithmic complexity routing for structured peer-to-peer networks. Its routing algorithm, however, does not provide an optimal trade-off between resources exploited (the size of the ‘finger table’) and performance (the average or worst-case number of hops to reach destination). Cordasco et al. showed that a finger table based o...

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...

The Horadam sequence is a generalization of the Fibonacci numbers in the complex plane, depending on a family of four complex parameters: two recurrence coefficients and two initial conditions. The necessary and sufficient periodicity conditions formulated in [1] are used to enumerate all Horadam sequences with a given period [2]. The geometry of periodic orbits is analyzed, where regular star-...

The Fibonacci lengths of the finite p-groups have been studied by R. Dikici and co-authors since 1992. All of the considered groups are of exponent p, and the lengths depend on the celebrated Wall number k(p). The study of p-groups of nilpotency class 3 and exponent p has been done in 2004 by R. Dikici as well. In this paper we study all of the p-groups of nilpotency class 3 and exponent p2. Th...

It is proved that the asymptotic average eccentricity and the asymptotic average degree of both Fibonacci cubes and Lucas cubes are (5 + √ 5)/10 and (5 − √ 5)/5, respectively. A new labeling of the leaves of Fibonacci trees is introduced and it is proved that the eccentricity of a vertex of a given Fibonacci cube is equal to the depth of the associated leaf in the corresponding Fibonacci tree. ...