Related papers: Integer Sequences and Output Arrays

We provide an efficient encoding of the natural numbers {0,1,2,3,...} as strings of nested parentheses {(),(()),(()()),((())),...}, or considered inversely, an efficient enumeration of such strings. The technique is based on the recursive…

In 1882 J.J. Sylvester already proved, that the number of different ways to partition a positive integer into consecutive positive integers exactly equals the number of odd divisors of that integer (see [1]). We will now develop an…

The Catalan numbers $C_k$ were first studied by Euler, in the context of enumerating triangulations of polygons $P_{k+2}$. Among the many generalizations of this sequence, the Fuss-Catalan numbers $C^{(d)}_k$ count enumerations of…

In this article, we study (102,000)-avoiding inversion sequences with a fixed number of distinct elements. By introducing simple H-paths, we derive the trivariate generating function for these inversion sequences with respect to their…

A sequence $(a_1, \ldots, a_n)$ of nonnegative integers is an {\em ascent sequence} if $a_0 =0$ and for all $i \geq 2$, $a_i$ is at most 1 plus the number of ascents in $(a_1, \ldots, a_{i-1})$. Ascent sequences were introduced by…

For all positive even integers $n$, graphs of order $n$ with degree sequence \begin{equation*} S_{n}:1,2,\dots,n/2,n/2,n/2+1,n/2+2,\dots,n-1 \end{equation*} naturally arose in the study of a labeling problem in \cite{IMO}. This fact…

In this short article, we will reconstruct the KP equation from Plucker relations and provide some generalizations on this topic. Additionally, in the final section, we define the discrete function $\tau$ in a similar manner, leading to the…

We define numbers of the type Oj(N) and Ej(N) and the corresponding integer sequences. We prove that these integer sequences, e.g., SO(N) and SE(N) correspond to the number of odd and even walks in complete graphs. We then prove that there…

The theory of sequences, supported by many SMT solvers, can model program data types including bounded arrays and lists. Sequences are parameterized by the element data type and provide operations such as accessing elements, concatenation,…

In this note we characterize all regular tetrahedra whose vertices in R^3 have integer coordinates. The main result is a consequence of the characterization of all equilateral triangles having integer coordinates contained in previous work.…

The Catalan number $C_n$ enumerates parenthesizations of $x_0*\dotsb*x_n$ where $*$ is a binary operation. We introduce the modular Catalan number $C_{k,n}$ to count equivalence classes of parenthesizations of $x_0*\dotsb*x_n$ when $*$…

In this paper we examine two ways of coding sequences in arithmetical theories. We investigate under what conditions they work. To be more precise, we study the creation of objects of a data-type that we call ur-strings, roughly sequences…

This paper gives a brief description of the author's database of integer sequences, now over 35 years old, together with a selection of a few of the most interesting sequences in the table. Many unsolved problems are mentioned.

The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the…

Inspired by the definition of modified ascent sequences, we introduce a new class of integer sequences called revised ascent sequences. These sequences are defined as Cayley permutations where each entry is a leftmost occurrence if and only…

Let the sequence S_m of nonnegative integers be generated by the following conditions: Set the first term a_0 = 0, and for all k \geq 0, let a_k+1 be the least integer greater than a_k such that no element of {a_0,...,a_k+1} is the average…

As a report of a teaching experience, we analyse Haskell programs computing two integer sequences: the Hamming sequence and the Ulam sequence. For both of them, we investigate two strategies of computation: the first is based on filtering…

Very recently, in [Das et al., J. Lond. Math. Soc., 2025], statistically characterized subgroups were studied for certain classes of non-arithmetic sequences. Subsequently, in [Das et al., Bull. Sci. Math., 2025], characterized subgroups…

Inversion sequences are integer sequences $(\sigma_1, \dots, \sigma_n)$ such that $0 \leqslant \sigma_i < i$ for all $1 \leqslant i \leqslant n$. The study of pattern-avoiding inversion sequences began in two independent articles by…

In this paper, we first give a new result characterizing the strongly connected digraphs with a diameter equal to that of their line digraphs. Then, we introduce the concepts of the inner diameter and inner radius of a digraph and study…