Related papers: On other two representations of the C-recursive in…
Let $(a_n), (b_n)$ be linear recursive sequences of integers with characteristic polynomials $A(X),B(X)\in \mathbb{Z}[X]$ respectively. Assume that $A(X)$ has a dominating and simple real root $\alpha$, while $B(X)$ has a pair of conjugate…
Two-term recurrence relations are supplied for indefinite integrals of functions that involve factors of the types ${P_2}^n$, ${P_3}^n$, ${P_4}^n$, ${P_1}^m {Q_1}^n$, $E_1 {P_1}^n$, ${P_1}^m {Q_2}^n$, $E_1 {P_2}^n$, ${P_2}^m {Q_2}^n$,…
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 propose a new definition of effective formulas for problems in enumerative combinatorics. We outline the proof of the fact that every linear recurrence sequence of integers has such a formula. It follows from a lower bound that can be…
Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence…
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…
A well-known generalisation of positional numeration systems is the case where the base is the residue class of $x$ modulo a given polynomial $f(x)$ with coefficients in (for example) the integers, and where we try to construct finite…
Let $\{U_n\}_{n \geq 0}$ and $\{V_m\}_{m \geq 0}$ be two linear recurrence sequences. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as differences $ U_n - V_m$. In…
In this paper, we find all integers $c$ having at least two representations as a difference between linear recurrent sequences. This problem is a pillai problem involving Padovan and Fibonacci sequence. The proof of our main theorem uses…
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 present linear forms with integer coefficients containing the Euler-Mascheroni and Euler-Gompertz constants. The forms are defined by four-terms recurrence relations. Asymptotics of the forms and their coefficients are obtained.
We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representating prime numbers by…
We prove recursive formulas involving sums of divisors and sums of triangular numbers and give a variety of identities relating arithmetic functions to divisor functions providing inductive identities for such arithmetic functions.
A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences…
Sums of the form $\sum_{N_m=q}^{n}{\cdots \sum_{N_1=q}^{N_2}{a_{(m);N_m}\cdots a_{(1);N_1}}}$ where the $a_{(k);N_k}$'s are same or distinct sequences appear quite often in mathematics. We will refer to them as recurrent sums. In this…
The notion of transducer integer sequences is considered through a series of examples. By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite automata encoding tree morphisms…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
For enumerative problems, i.e. computable functions f from N to Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f(n) in the number of steps that is polynomial in the combined size of the input n and…
In this note, we show the existence of integer sequences of lengths at least 3 (except 7) such that for every integer in position $i\equiv 1\pmod{4}$ (respectively position $j\equiv 3\pmod{4}$), counting from left to right, the sum of the…
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…