Related papers: On Base 3/2 and its Sequences
We consider the avoidance of patterns in inversion sequences that relate sorting via sorting machines including data structures such as pop stacks and stacks. Such machines have been studied under a variety of additional constraints and…
In this article we charaterize the primes Fibonacci numbers of the form $x^2 +ry^2$, where $r = 1,$ $r$ is a prime positive integer number or r is a power of a prime positive integer, using techniques of combinatorics and numbers theory. We…
In this paper 101 new integer sequences, sub-sequences, and sequences of sequences, together with related unsolved problems and conjectures, are presented. Also, definitions, examples, solved or open questions, and references for each…
Understanding the distribution of digits in the expansions of perfect powers in different bases is difficult. Rather than consider the asymptotic digit distributions, we consider the base-10 digits of a restricted sequence of powers of two.…
This note is devoted to study the recurrent numerical sequence defined by: $a_0 = 0$, $a_n = \frac{n}{2} a_{n - 1} + (n - 1)!$ ($\forall n \geq 1$). Although, it is immediate that ${(a_n)}_n$ is constituted of rational numbers with…
We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.
We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…
This work represents an in-depth study of the structural behavior of the Collatz sequences. We consider a finite arithmetic progression with a common difference is 2 and the number of terms in the sequence is equal to 2^n . After, we…
In a permutation sequence built by means of sub permutations the transition between successive permutations are subject to a set of n(n - 1)/2 rules that group into n - 1 matrices with a high degree of regularity. By means of these rules…
In this paper, we provide properties and applications of some special integer sequences. We generalize and give some properties of Pisano period. Moreover, we provide a new application in Cryptography and applications of some quaternion…
We give a survey of the foundations of statistical queries and their many applications to other areas. We introduce the model, give the main definitions, and we explore the fundamental theory statistical queries and how how it connects to…
In this paper, we show that the concatenation of the Fibonacci sequence is \textit{normal} in base $10$, meaning every string of a given length, $k$, occurs as frequently as every other string of length $k$ (there are as many $1$'s as $2$'s…
We give some results and conjectures about recurrence relations for certain sequences of binomial sums.
Let a tribonacci sequence be a sequence of integers satisfying $a_k=a_{k-1}+a_{k-2}+a_{k-3}$ for all $k\ge 4$. For any positive integers $k$ and $n$, denote by $f_k(n)$ the number of tribonacci sequences with $a_1, a_2, a_3>0$ and with…
Let $S_b(n)$ denote the sum of the squares of the digits of the positive integer $n$ in base $b\geq2$. It is well-known that the sequence of iterates of $S_b(n)$ terminates in a fixed point or enters a cycle. Let $N=2n-1$, $n\geq2$. It is…
Wall published a paper in 1960 on the Fibonacci sequence where he derived many results concerning the period and prime power divisibility modulo m. His periodicity results have been generalized to second order linear recurrences. Here we…
Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials…
Simple methods permit to generalize the concepts of iteration and of recursive processes. We shall see briefly on several examples what these methods generate. In additive sequences, we shall encounter not only the golden or the silver…
The divisibility of truncated binomial series by their exponent n is analyzed. Divisibility is shown to depends on the divisibility characteristics of the integers constituting the binomials. Series division by the highest possible powers…
A chain is an ordering of the integers 1 to n such that adjacent pairs have sums of a particular form, such as squares, cubes, triangular numbers, pentagonal numbers, or Fibonacci numbers. For example 4 1 2 3 5 form a Fibonacci chain while…