Related papers: The structure of base phi expansions

Let $H$ be the Iwahori--Hecke algebra corresponding to any Coxeter group. Deodhar's defect statistic [Geom. Dedicata 36, (1990) pp.95--119] allows one to expand products of simple Kazhdan--Lusztig basis elements of $H$ in the natural basis…

We introduce two families of transcendental numbers which we call finite factorial (FF) and partially finite factorial (PFF) numbers respectively, with the former one being subfamily of the latter one. These numbers arise naturally from…

For an integer $n\geq 1$, consider the $n$-th metallic number $\phi_n=\frac{n+\sqrt{n^2+4}}{2}$ (e.g. $\phi_1$ is the golden number) and denote by $[\phi_n]_q$ its $q$-deformation in the sense of S. Morier-Genoud and V. Ovsienko. This is an…

Recently, there has been a sharp rise of interest in properties of digits primes. Here we study yet another question of this kind. Namely, we fix an integer base $g \ge 2$ and then for every infinite sequence $${\mathcal D} =…

A generalization of the well-known Fibonacci sequence is the $k$-Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,0, \ldots, 1$, and each term afterwards is the sum of the preceding $k$ terms.…

Let $f(x) \in \mathbb{Z}[x]$. Set $f_{0}(x) = x$ and, for $n \geq 1$, define $f_{n}(x)$ $=$ $f(f_{n-1}(x))$. We describe several infinite families of polynomials for which the infinite product \prod_{n=0}^{\infty} (1 + \frac{1}{f_{n}(x)})…

Zeckendorf's Theorem implies that the Fibonacci number $F_n$ is the smallest positive integer that cannot be written as a sum of non-consecutive previous Fibonacci numbers. Catral et al. studied a variation of the Fibonacci sequence, the…

The paper contains enumerative combinatorics for positive braids, square free braids, and simple braids, emphasizing connections with classical Fibonacci sequence. The simple subgraph of the Cayley graph of the braid group is analyzed in…

This article is about Pi Formulas, infinite series of fractions which sum to multiples of Pi. Each such one can be associated with a unique set $S_k$ of rough numbers, where $k$ is a prime number. Given $S_k$ for any prime $k$, the set…

We study the $k$-Bonacci word over the infinite alphabet $\mathbb{N}$. Since the alphabet is infinite, the usual factor complexity is infinite and does not provide any information. We therefore investigate factor occurrence statistics in…

For alternate Cantor real base numeration systems we generalize the result of Frougny and~Solomyak on~arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness…

There are two basic number sequences which play a major role in the prime number distribution. The first Number Sequence SQ1 contains all prime numbers of the form 6n+5 and the second Number Sequence SQ2 contains all prime numbers of the…

An element w of the extension E of degree n over the finite field F=GF(q) is called free over F if {w, w^q,...,w^{q^{n-1}}} is a (normal) basis of E/F. The Primitive Normal Basis Theorem, first established in full by Lenstra and Schoof…

We define Poisson genericity for infinite sequences in any finite or countable alphabet with an invariant exponentially-mixing probability measure. A sequence is Poisson generic if the number of occurrences of blocks of symbols…

Let $\theta$ and $\rho$ be real numbers with $0 \le \theta, \rho < 1$ and $\theta$ irrational. We show that the Hecke-Mahler series $$ F_{\theta, \rho} (z_1, z_2) = \sum_{k_1 \ge 1} \, \sum_{k_2 = 1}^{\lfloor k_1 \theta + \rho \rfloor} \,…

We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n^4 mathematical operations. The…

Answering a question of Erd\H{o}s and Graham, we show that the double exponential growth condition $\limsup_{n\to\infty}a_n^{1/\phi^n}=\infty$ for a strictly increasing sequence of positive integers $\{a_n\}_{n=1}^\infty$ is sufficient for…

This paper investigates the natural density and structural relationships within Fibonacci words, the density of a Fibonacci word is $\operatorname{DF}(F_k)=n/(n+m),$ where $m$ denote the number of zeros in a Fibonacci word and $n$ denote…

A famous theorem of Szemer\'edi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a…

Let $f(z)=\sum_{n=1}^{\infty}a(n) e^{2\pi i nz}$ be a normalized Hecke eigenform in $S_{2k}^{\text{new}}(\Gamma_0(N))$ with integer Fourier coefficients. We prove that there exists a constant $C(f)>0$ such that any integer is a sum of at…