The paper investigates relationship between algebraic expressions and Fibonacci graphs (which give a generic example of non-series-parallel graphs). We propose the uniform generalized decomposition method for constructing Fibonacci graph expressions. On every step this method divides the graph of size n into k parts of the same size. We prove that to reach the smallest possible length of the co...

We study the quasiperiodic behavior of electrons in a Fibonacci magnetic superlattice, produced on a two-dimensional electron gas (2DEG) modulated by magnetic field stripes arranged in a Fibonacci sequence. A rich (multifractal) resonant structure of transmission and conductance spectra is revealed. Although electron motion in a magnetically modulated 2DEG is inherently a two dimensional proces...

Hypergeometric functions are an important tool in many branches of pure and applied mathematics, and they encompass most special functions, including the Chebyshev polynomials. There are also well-known connections between Chebyshev polynomials and sequences of numbers and polynomials related to Fibonacci numbers. However, to my knowledge and with one small exception, direct connections between...

we introduce a matricial toeplitz transform and prove that the toeplitz transform of a second order recurrence sequence is another second order recurrence sequence. we investigate the injectivity of this transform and show how this distinguishes the fibonacci sequence among other recurrence sequences. we then obtain new fibonacci identities as an application of our transform.

We provide explicit formulas for the maximum number qk(n) of disjoint subgraphs isomorphic to the k-dimensional hypercube in the n-dimensional Fibonacci cube Γn for small k, and prove that the limit of the ratio of such cubes to the number of vertices in Γn is 1 2k for arbitrary k. This settles a conjecture of Gravier, Mollard, Špacapan and Zemljič about the limiting behavior of qk(n).

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 1985, Simion and Schmidt showed that |Sn(τ3)|, the cardinality of the set of all length n permutations avoiding the patterns τ3 = {123, 213, 132} is the Fibonacci numbers, fn+1. They also developed a constructive bijection between the set of all binary strings with no two consecutive ones and Sn(τ3). In May 2004, Egge and Mansour generalized this SimionSchmidt counting result and showed that...

The skew of a binary string is the difference between the number of zeroes and the number of ones, while the length of the string is the sum of these two numbers. We consider certain suffixes of the lexicographicallyleast de Bruijn sequence at natural breakpoints of the binary string. We show that the skew and length of these suffixes are enumerated by sequences generalizing the Fibonacci and L...

The non-repetitive complexity function of an infinite word x (first defined by Moothathu) is the function of n that counts the number of distinct factors of length n that appear at the beginning of x prior to the first repetition of a length-n factor. We examine general properties of the non-repetitive complexity function, as well as obtain formulas for the non-repetitive complexity of the Thue...