N. J. A. Sloane
This article gives a brief introduction to the On-Line Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other information.
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Partial Dyck path interpretation for three sequences in the Encyclopedia of Integer Sequences
本文利用生成函数与核方法研究 Dyck 路径的奇数长度下降及其变体，为《整数序列百科全书》中三个相关序列给出部分 Dyck 路径解释。
The recent history of The On-Line Encyclopedia of Integer Sequences (or OEIS), describing developments since 2009, and discussing recent sequences involving interesting unsolved problems and in many cases spectacular illustrations. These include: Peaceable Queens, circles in the plane, the earliest cube-free binary sequence, the EKG and Yellowstone permutations, other lexicographically earliest sequences, iteration of number-theoretic functions, home primes and power trains, a memorable prime, a missing prime, Post's tag system, and coordination sequences.
Stephan (Prove or Disprove 100 Conjectures from the OES, arXiv:math/0409509v4 [math.CO])enumerates a number of conjectures regarding integer sequences contained in Sloane's On-line Encyclopedia of Integer Sequences (N. J. A. Sloane, editor, The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org). In this paper, we prove two of these conjectures.
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. Then we use this characterization to point out some corollaries. The number of such tetrahedra whose vertices are in the finite set {0,1,2,...,n}^3, n in N, is related to the sequence A103158 in the Online Encyclopedia of Integer Sequences.
Sequences whose terms are equal to the number of functions with specified properties are considered. Properties are based on the notion of derangements in a more general sense. Several sequences which generalize the standard notion of derangements are thus obtained. These sequences generate a number of integer sequences from the well-known Sloane's encyclopedia.
We show that sequences A026737 and A111279 in The On-Line Encyclopedia of Integer Sequences are the same by giving a bijection between two classes of Grand Schroder paths.
Until 1973 there was no database of integer sequences. Someone coming across the sequence 1, 2, 4, 9, 21, 51, 127,... would have had no way of discovering that it had been studied since 1870 (today these are called the Motzkin numbers, and form entry A001006 in the database). Everything changed in 1973 with the publication of "A Handbook of Integer Sequences", which listed 2372 entries. This report describes the fifty-year evolution of the database from the "Handbook" to its present form as "The On-Line Encyclopedia of Integer Sequences" (or OEIS), which contains 360,000 entries, receives a mi…
Recurrence Relations for Some Integer Sequences Related to Ward Numbers
本文研究了与Ward数相关的整数序列，给出了Ward-Lah数、变型Ward数和二项式Ward数等多种序列的三角递推关系、水平递推关系、生成函数及高阶递推关系。
In this expository article we collect the integer sequences that count several different types of matrices over finite fields and provide references to the Online Encyclopedia of Integer Sequences (OEIS). Section 1 contains the sequences, their generating functions, and examples. Section 2 contains the proofs of the formulas for the coefficients and the generating functions of those sequences if the proofs are not easily available in the literature. The cycle index for matrices is an essential ingredient in most of the derivations.
In his July 1974 Scientific American column, Martin Gardner mentioned the Handbook of Integer Sequences, which then contained 2372 sequences. Today the On-Line Encyclopedia of Integer Sequences (the OEIS) contains 140000 sequences. This paper discusses eight of them, suggested by the theme of the Eighth Gathering For Gardner: they are all infinite, and all 'ateful in one way or another. Each one is connected with an unsolved problem. The sequences are related to: hateful numbers, Angelini's 1995 puzzle, the persistence of a number, Alekseyev's 123 sequence, the curling number conjecture, Quet'…
An introduction to the On-Line Encyclopedia of Integer Sequences (or OEIS, https://oeis.org) for graduate students in mathematics
We give the connection between three polynomials that generate triangles in The On-Line Encyclopedia of Integer Sequences (A123192, A137396 and A300453). We show that they are related with the bracket polynomial for the (2,n)-torus knot
We give an in depth analysis of an algorithm that generates permutations of the natural numbers introduced by Clark Kimberling in the On Line Encyclopedia of Integer Sequences. It turns out that the examples of such permutations in OEIS are completely determined by 3-automatic sequences.
On the Number of Regular Integers Modulo $n$ and Its Significance for Cryptography
该研究给出了模n正则整数计数的Morgado公式的四种组合证明，明确其对应OEIS的A055653序列，并将其与多素数多幂次的RSA密码系统推广方案关联，体现其密码学应用价值。
We look at a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).
Some Fibonacci-Related Sequences
本文研究了递归定义的序列A105774的性质，利用Fibonacci表示、有限自动机和Walnut定理证明器作为主要工具，并证明了关于同步序列的两个新结果。
Sequence A000975 in the Online Encyclopedia of Integer Sequences (OEIS) starts out 1, 2, 5, 10, 21, 42, 85, ... . As of July 1, 2016, the description in the OEIS lists several characterizations of this sequence and numerous examples of instances where this sequence occurs. It also presents a "not yet proved" result, a conjecture, and an unanswered question concerning this sequence. In this paper we show that all of these proposed results are in fact true.
Let $β$ be any permutation on $n$ symbols and let $c(k, β)$ be the number of permutations that $k$-commute with $β$. The cycle type of a permutation $β$ is a vector $(c_1, \dots, c_n)$ such that $β$ has exactly $c_i$ cycles of length $i$ in its disjoint cycle factorization. In this article we obtain formulas for $c(k, β)$, for some cycle types. We also express these formulas in terms of integer sequences as given in "The On-line Encyclopedia of Integer Sequences" (OEIS). For some of these sequences we obtain either new interpretations or relationships with sequences in the OEIS database.
We have exhaustively enumerated all simple, connected graphs of a finite order and have computed a selection of invariants over this set. Integer sequences were constructed from these invariants and checked against the Online Encyclopedia of Integer Sequences (OEIS). 141 new sequences were added and 6 sequences were appended or corrected. From the graph database, we were able to programmatically suggest relationships among the invariants. It will be shown that we can readily visualize any sequence of graphs with a given criteria. The code has been released as an open-source framework for furth…
Generating functions of lattice paths
本文回顾格路径的主要类型，并利用 Riordan 数组方法统一、澄清和简化各类格路径的生成函数及相关整数序列。
Edge-magic total labeling was introduced by \cite{GS}. The number of edge-magic solutions for cycles have been explored in \cite{BS}. This sequence is mentioned in On Line Encyclopedia of Integer Sequences (OEIS) \cite{oeis}. In this short note, we enumerate the number of unique edge-magic total labelings on Path $P_n$
We discuss an equivalence relation on the set of square binary matrices with the same number of 1's in each row and each column. Each binary matrix is represented using ordered n-tuples of natural numbers. We give a few starting values of integer sequences related to the discussed problem. The obtained sequences are new and they are not described in the On-Line Encyclopedia of Integer Sequences (OEIS). We show a relationship between some particular values of the parameters and the Fibonacci sequence.
Hardinian Arrays
本文证明了Kauers和Koutschan关于Hardin序列的递推猜想及OEIS中的旧猜想，并提出了关于序列渐近性的新猜想。
Counting appearances of integers in sets of arithmetic progressions
通过行列式方法，研究了整数在由k个不同素数生成的等差数列余数类中的出现次数计数问题。
The sequence A120986 in the Encyclopedia of Integer Sequences counts ternary trees according to the number of nodes and the number of middle edges. Using a certain substition, the underlying cubic equation can be factored. This leads to an extension of the concept of $(3/2)$-ary trees, introduced by Knuth in his christmas lecture from 2014.
Proofs of several OEIS conjectures on determinants and permanents
利用行和列操作、块行列式公式、柯西行列式、Sylvester行列式定理和LU分解，证明了在线整数序列百科中关于特殊矩阵（如Toeplitz矩阵、交叉矩阵、Kronecker幂矩阵及差幂矩阵）的行列式和积和式的几个猜想，并得到了若干相关整数序列的闭式公式。
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 (length and prefix preserving transformations of words). Transducer integer sequences are related to the notion of self-similar groups and semigroups, as well as to the notion of automatic sequences.
Some inner metric parameters of a digraph: Iterated line digraphs and integer sequences
本文研究有向图的内直径与内半径在迭代线有向图中的演化规律，提出刻画相关整数序列的方法，并应用于 Kautz 有向图、无平方有向图及 De Bruijn 子图，发现若干新整数序列。
We consider the sequence of integers whose $n$th term has base-$p$ expansion given by the $n$th row of Pascal's triangle modulo $p$ (where $p$ is a prime number). We first present and generalize well-known relations concerning this sequence. Then, with the great help of Sloane's On-Line Encyclopedia of Integer Sequences, we show that it appears naturally as a subsequence of a $2$-regular sequence. Its study provides interesting relations and surprisingly involves odious and evil numbers, Nim-sum and even Gray codes. Furthermore, we examine similar sequences emerging from prime numbers involvin…
The generating function of A348410 in OEIS using the diagonal method and another sequence (A001008) from OEIS
本文利用Hautus和Klarner的方法及软件Gfun，推导出序列A348410的生成函数满足的代数方程，并借助广义二项级数得到另一个恒等式。
Convolutive sequences, I: Through the lens of integer partition functions
本文研究了满足特定卷积性质的序列，通过组合学方法和生成函数操作，探讨了整数划分函数与eta积的关系，并给出了相关序列的实例。
Partitions of [n]={1,2,...,n} into sets of lists are counted by sequence number A000262 in the On-Line Encyclopedia of Integer Sequences. They are somewhat less numerous than partitions of [n] into lists of sets, A000670. Here we observe that the former are actually equinumerous with partitions of [n] into lists of *noncrossing* sets and give a bijective proof. We show that partitions of [n] into sets of noncrossing lists are counted by A088368 and generalize this result to introduce a transform on integer sequences that we dub the "noncrossing partition" transform. We also derive recurrence r…
On the Number of Exceptional Pairs Over a Class of Nakayama Algebras
本文利用组合与同调方法，为若干Nakayama代数及根平方零代数的Auslander代数推导出例外对数量的显式公式，并揭示其与OEIS整数序列的联系。
We present an algorithm to find invariant poynomial transformations of integer sequences, using the classical invariant theory approach.
In this note, we explore two families of sequences associated to a suitable integer sequence: the gap-sum sequence and the gap-product sequence. These are the sums and the products of consecutive numbers not in the original sequence. We give closed forms for both, in terms of the original sequence, and in the case of Horadam sequences, we find the generating function of the gap-sum sequence. For some elementary sequences, we indicate that the gap-product sequences are given by the Fuss-Catalan-Raney numbers.
The first author recently introduced an integer sequence now numbered A355519 in OEIS. This sequence arose from counting bracket tournaments; its study evokes the analysis of the Catalan triangle (sequence A009766 in OEIS) and the related Catalan numbers, sequence A000108 in OEIS. We therefore introduce a general construction that places these sequences on the same footing and suggests many new integer sequences. We provide code for performing this construction and a lengthy list of examples. This construction determines a function from input sequences to output sequences. Some of the resultin…
Effective MC-finiteness
本文针对MC有限序列中函数F:(m,n)↦aₙ mod m的可计算性问题，讨论了其可有效计算的情况，此前相关研究仅给出OEIS中的MC有限序列例子未涉及该可计算性。
A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.
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.
Stern's diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit representation of integers is an alternative representation of integers with much use in efficient computation, coding theory and cryptography. We link these two ideas here, showing that the number of $i$-bit binary signed-digit representations of an integer $n$ with $n<2^i$ is the $(2^i-n)^\text{th}$ element in Stern's diatomic sequence. This correspondence makes the vast range of results known for Stern's diatomic sequence available for consideration in the s…
Let A be a set of integers dense in a finite interval. We establish upper and lower bounds for the longest regularly-spaced and convex subsets of A and of A-A.
When my "Handbook of Integer Sequences" came out in 1973, Philip Morrison gave it an enthusiastic review in the Scientific American and Martin Gardner was kind enough to say in his Mathematical Games column that "every recreational mathematician should buy a copy forthwith." That book contained 2372 sequences. Today the "On-Line Encyclopedia of Integer Sequences" contains 117000 sequences. This paper will describe seven that I find especially interesting. These are the EKG sequence, Gijswijt's sequence, a numerical analog of Aronson's sequence, approximate squaring, the integrality of n-th roo…
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 deduced from the Skolem-Mahler-Lech theorem and the Subspace Theorem. We will give details of this deduction that is due to P. Corvaja in the full version of this extended abstract.
On Some Properties of Matrices with Entries Defined by Products of $k$-Fibonacci and $k$-Lucas Numbers
本文研究了由k-斐波那契数和k-卢卡斯数乘积定义的矩阵家族，推导了多个矩阵不变量的显式公式，并探讨了与整数序列的关系。
How long can $k$-Göbel sequences remain integers?
受漫画启发，研究$k$-Göbel序列的整数性，证明对任意$k\geq2$，前19项（$n\leq18$）均为整数。
In this tribute to Neil Sloane, we revisit the first sequence in the On-Line Encyclopedia of Integer Sequences, sequence A435 (1, 8, 78, 944, 13800, 237432, 4708144, 105822432, ...), that he encountered when he was a graduate student, and when normalized gives the average total height of rooted labeled trees. We state rigorously-computed explicit expressions for the first twelve moments of the random variable `total height' on rooted labeled trees, and pledge to donate to the OEIS 100 dollars in honor of the first to find an explicit expression for the probability density function of the limit…
Number of partitions of modular integers (with an Appendix by P. Deligne)
本研究给出模n整数环中k阶子集和为s的计数公式，通过构造满足克罗内克积性质的矩阵序列M(k)推导闭式解，关联项链数与Lyndon词，结合拉马努金和理论完成证明，覆盖大量已知整数序列。
The following general idea looks crazy. What if another integer sequence follows each integer sequence like a shadow? I will demonstrate that this is indeed the case, perhaps not for every integer sequence, but for many of them.
We propose several procedures for creating new families of integer sequences based on the method of Cantor diagonalization. Then we modify and generalize this method. The paper includes explicit formulas for most proposed families of integer sequences.
Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between integer partitions with kinds and finite differences of the coefficients of generalized Galois numbers is established.
Stern's diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit (BSD) representation of integers is used widely in efficient computation, coding theory and other applications. We link these two objects, showing that the number of $i$-bit binary signed-digit representations of an integer $n<2^i$ is the $(2^i-n)^\text{th}$ element in Stern's diatomic sequence. This correspondence makes the vast range of results known about the Stern diatomic sequence available for consideration in the study of binary signed-digit integers, an…
We use the concept of the half of a lower-triangular matrix to define a transformation on integer sequences. We explore the properties of this transformation, including in some cases a study of the Hankel transform of the transformed sequences. Starting from simple sequences with elementary rational generating functions, we obtain many sequences of combinatorial significance. We make extensive use of techniques drawn from the theory of Riordan arrays.
On the length over which $k$-Göbel sequences remain integers
本文证明$k$-Göbel序列保持整数的长度$N_k$无界，即存在任意长的初始整数段。
Sequence Factory: Families of Integer Sequences Generated by Gödel Implication
本文研究哥德尔蕴含生成的整数序列族，证明根对生成函数初始系数可唯一确定无序根值对，编制了m=4至10的序列工厂目录，为整数序列分析及与OEIS对比提供可复现数据。
Integer Sequences: Irregular Arrays and Intra-Block Permutations
本文研究不规则数组的整数序列，给出编号显式公式，推广康托尔对角线方法，并定义块内置换及广义不情愿序列。
To partition a sequence of n integers into subsets with prescribed sums is an NP-hard problem in general. In this paper we present an efficient solution for the homogeneous version of this problem; i.e. where the elements in each subset add up to the same sum.
Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1, \ldots, x_n\}$ of integers where $x_1<\cdots<x_n$, let the {\it gap sequence} of this set be the nondecreasing sequence $d_1, \ldots, d_{n-1}$ where $\{d_1, \ldots, d_{n-1}\}$ equals $\{x_{i+1}-x_i:i\in\{1,\ldots, n-1\}\}$ as a multiset. This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets …
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 is a unique family of graphs which have exactly the same sequence of odd walks between connected nodes and of even walks between pairs of nodes at distance two, respectively. These graphs are obtained as the Kronecker product. We show that they are the incidence graphs of block designs, are distance-regular and Ramanujan graphs.
Attractors of sequences coding $β$-integers
研究编码β-整数的序列的吸引子问题，通过对简单帕里序列及特殊二元非简单帕里序列的研究，证实关于吸引子最小性的猜想，给出特定序列前缀的最小吸引子。
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 of m - 1 distinct other elements. Szekeres gave a closed-form description of S_3 in 1936, and Layman provided a similar description for S_4 in 1999. We first find closed forms for some similar greedy sequences that avoid averages in terms not all the same. Then, we extend the closed-form description of S_m from the known cases when m = 3 and m = 4 to any int…
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