Related papers: The Fibonacci Sequence is Normal Base 10

We prove connections between Zeckendorf decompositions and Benford's law. Recall that if we define the Fibonacci numbers by $F_1 = 1, F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$, every positive integer can be written uniquely as a sum of…

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 lexicographically-least de Bruijn…

We colour the Fibonacci sequence by suitable constant gap sequences to provide an upper bound on the asymptotic repetitive threshold of $d$-ary balanced sequences. The bound is attained for $d=2, 4$ and $8$ and we conjecture that it happens…

We introduce a class of stochastic integer sequences. In these sequences, every element is a sum of two previous elements, at least one of which is chosen randomly. The interplay between randomness and memory underlying these sequences…

We study a game in which one keeps flipping a coin until a given finite string of heads and tails occurs. We find the expected number of coin flips to end the game when the ending string consists of at most four maximal runs of heads or…

We introduce the concept of Minkowski normality, a different type of normality for the regular continued fraction expansion. We use the ordering \[ \frac{1}{2},\quad \frac{1}{3}, \frac{2}{3},\quad \frac{1}{4}, \frac{3}{4},\frac{2}{5},…

The random Fibonacci chain is a generalisation of the classical Fibonacci substitution and is defined as the rule mapping $0\mapsto 1$ and $1 \mapsto 01$ with probability $p$ and $1 \mapsto 10$ with probability $1-p$ for $0<p<1$ and where…

The Collatz sequence for a given natural number $N$ is generated by repeatedly applying the map $N$ $\rightarrow$ $3N+1$ if $N$ is odd and $N$ $\rightarrow$ $N/2$ if $N$ is even. One elusive open problem in Mathematics is whether all such…

Let ${\cal P}$ be the set of palindromes occurring in the Fibonacci sequence. In this note, we establish three structures of $\mathcal{P}$ and and discuss their properties: cylinder structure, chain structure and recursive structure. Using…

We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such…

A base-b junction number u has the property that there are at least two ways to write it as u = v + s(v), where s(v) is the sum of the digits in the expansion of the number v in base b. For the base-10 case, Kaprekar in the 1950's and…

We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…

Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with…

Solutions to the random Fibonacci recurrence x_{n+1}=x_{n} + or - Bx_{n-1} decrease (increase) exponentially, x_{n} = exp(lambda n), for sufficiently small (large) B. In the limits B --> 0 and B --> infinity, we expand the Lyapunov exponent…

We investigate the problem of finding integers $k$ such that appending any number of copies of the base-ten digit $d$ to $k$ yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits…

It is well known that if $0.a_1a_2a_3\dots$ is the base-$b$ expansion of a number normal to base-$b$, then the numbers $0.a_ka_{m+k}a_{2m+k}\dots$ for $m\ge 2$, $k\ge 1$ are all normal to base-$b$ as well. In contrast, given a continued…

In 1999, De Simone and K\"{o}rner conjectured that every graph without induced $C_5,C_7,\overline{C}_7$ contains a clique cover $\mathcal C$ and a stable set cover $\mathcal I$ such that every clique in $\mathcal C$ and every stable set in…

For any real-valued $k > 1$, we consider the tree rooted at 0, where each positive integer $n$ has parent $\lfloor\frac{n}{k}\rfloor$. The average number of children per node is $k$, thus this definition gives a natural way to extend…

This work presents a mathematical model that establishes an interesting connection between nucleotide frequencies in human single-stranded DNA and the famous Fibonacci's numbers. The model relies on two assumptions. First, Chargaff's second…

In this paper, we define the bi-periodic Fibonacci matrix sequence that represent bi-periodic Fibonacci numbers. Then, we investigate generating function, Binet formula and summations of bi-periodic Fibonacci matrix sequence. After that, we…