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N. J. A. Sloane

Publications and source records attributed to N. J. A. Sloane.

At least 19 recordsLinked to original sources
arXiv2024

The Comma Sequence: A Simple Sequence With Bizarre Properties

The ``comma sequence'' starts with 1 and is defined by the property that if k and k' are consecutive terms, the two-digit number formed from the last digit of k and the first digit of k' is equal to the difference k'-k. If there is more than one such k', choose the smallest, but if there is no such k' the sequence terminates. The sequence begins 1, 12, 35, 94, 135, ... and, surprisingly, ends at term 2137453, which is 99999945. The paper analyzes the sequence and its generalizations to other starting values and other bases. A slight change in the rules allows infinitely long comma sequences to exist.

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arXiv2023

On Dissecting Polygons into Rectangles

What is the smallest number of pieces that you can cut an n-sided regular polygon into so that the pieces can be rearranged to form a rectangle? Call it r(n). The rectangle may have any proportions you wish, as long as it is a rectangle. The rules are the same as for the classical problem where the rearranged pieces must form a square. Let s(n) denote the minimum number of pieces for that problem. For both problems the pieces may be turned over and the cuts must be simple curves. The conjectured values of s(n), 3 <= n <= 12, are 4, 1, 6, 5, 7, 5, 9, 7, 10, 6. However, only s(4)=1 is known for certain. The problem of finding r(n) has received less attention. In this paper we give constructions showing that r(n) for 3 <= n <= 12 is at most 2, 1, 4, 3, 5, 4, 7, 4, 9, 5, improving on the bounds for s(n) in every case except n=4. For the 10-gon our construction uses three fewer pieces than the bound for s(10). Only r(3) and r(4) are known for certain. We also briefly discuss q(n), the minimum number of pieces needed to dissect a regular n-gon into a monotile.

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arXiv2023

"A Handbook of Integer Sequences" Fifty Years Later

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 million visits a day, and has been cited 10,000 times, often with a comment saying "discovered thanks to the OEIS".

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arXiv2022

The Binary Two-Up Sequence

The Binary Two-Up Sequence is the lexicographically earliest sequence of distinct nonnegative integers with the property that the binary expansion of the n-th term has no 1-bits in common with any of the previous floor(n/2) terms. We show that the sequence can be decomposed into ``atoms'', which are sequences of 4, 6, or 8 numbers whose binary expansions match certain patterns, and that the sequence is the limiting form of a certain ``word'' involving the atoms. This leads to a fairly explicit formula for the terms, and in particular establishes the conjecture that every nonzero term is the sum of at most two powers of 2.

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arXiv2022

On Kaprekar's Junction Numbers

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 1960's studied the problem of finding K(n), the smallest u such that the equation u=v+s(v) has exactly n solutions. He gave the values K(2)=101, K(3)=10^13+1, and conjectured that K(4)=10^24+102. In 1966 Narasinga Rao gave the upper bound 10^1111111111124+102 for K(5), as well as upper bounds for K(6), K(7), K(8), and K(16). In the present work, we derive a set of recurrences which determine K(n) for any base b and in particular imply that these conjectured values of K(n) are correct. The key to our approach is an apparently new recurrence for F(u), the number of solutions to u=v+s(v). We illustrate our method by computing the values of K(n) for n <= 16 and bases b <= 10, and show that for each base K(n) grows as a tower of height proportional to log_2(n). Rather surprisingly, the values of K(n) for the base-5 problem are determined by the classical Thue-Morse sequence, which leads us to define generalized Thue-Morse sequences for other bases.

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arXiv2021

Three Cousins of Recaman's Sequence

Although 10^230 terms of Recaman's sequence have been computed, it remains a mystery. Here three distant cousins of that sequence are described, one of which is also mysterious. (i) {A(n), n >= 3} is defined as follows. Start with n, and add n+1, n+2, n+3, ..., stopping after adding n+k if the sum n + (n+1) + ... + (n+k) is divisible by n+k+1. Then A(n)=k. We determine A(n) and show that A(n) <= n^2 - 2n - 1. (ii) {B(n), n >= 1} is a multiplicative analog of {A(n)}. Start with n, and successively multiply by n+1, n+2, ..., stopping after multiplying by n+k if the product n(n+1)...(n+k) is divisible by n+k+1. Then B(n)=k. We conjecture that log^2 B(n) = (1/2 + o(1)) log n loglog n. (iii) The third sequence, {C(n), n >= 1}, is the most interesting, because the most mysterious. Concatenate the decimal digits of n, n+1, n+2, ... until the concatenation n||n+1||...||n+k is divisible by n+k+1. Then C(n)=k. If no such k exists we set C(n)=-1. We have found k for all n <= 1000 except for two cases. Some of the numbers involved are quite large. For example, C(92) = 218128159460, and the concatenation 92||93||...||(92+C(92)) is a number with about 2*10^12 digits. We have only a probabilistic argument that such a k exists for all n.

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arXiv2020

Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids

A survey of enumeration problems arising from the study of graphs formed when the edges of a polygon are marked with evenly spaced points and every pair of points is joined by a line. A few of these problems have been solved, a classical example being the the graph K_n formed when all pairs of vertices of a regular n-gon are joined by chords, which was analyzed by Poonen and Rubinstein in 1998. Most of these problems are unsolved, however, and this two-part article provides data from a number of such problems as well as colored illustrations, which are often reminiscent of stained glass windows. The polygons considered include rectangles, hollow rectangles (or frames), triangles, pentagons, pentagrams, crosses, etc., as well as figures formed by drawing semicircles joining equally-spaced points on a line. %The paper ends with a brief discussion of the problem of how to %design aesthetically pleasing colorings for these graphs. This first part discusses rectangular grids. The 1 X n grids, or equally the graphs K_{n+1,n+1}, were studied by Legendre and Griffiths, and here we investigate the number of cells with a given number of edges and the number of nodes with a given degree. We have only partial results for the m X n rectangles, including upper bounds on the numbers of nodes and cells.

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arXiv2019

Queens in exile: non-attacking queens on infinite chess boards

Number the cells of a (possibly infinite) chessboard in some way with the numbers 0, 1, 2, ... Consider the cells in order, placing a queen in a cell if and only if it would not attack any earlier queen. The problem is to determine the positions of the queens. We study the problem for a doubly-infinite chessboard of size Z x Z numbered along a square spiral, and an infinite single-quadrant chessboard (of size N x N) numbered along antidiagonals. We give a fairly complete solution in the first case, based on the Tribonacci word. There are connections with combinatorial games.

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arXiv2019

"Choix de Bruxelles": A New Operation on Positive Integers

The "Choix de Bruxelles" operation replaces a positive integer n by any of the numbers that can be obtained by halving or doubling a substring of the decimal representation of n. For example, 16 can become any of 16, 26, 13, 112, 8, or 32. We investigate the properties of this interesting operation and its iterates.

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arXiv2019

A Coloring Book Approach to Finding Coordination Sequences

An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. We illustrate the method by first applying it to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual-3^2.4.3.4) tiling. The coordination sequence for a tetravalent vertex turns out, surprisingly, to be 1, 4, 8 ,12, 16, ..., the same as for a vertex in the familiar square (or 4^4) tiling. We thought that such a simple fact should have a simple proof, and this article is the result. We also apply the method to obtain coordination sequences for the 3^2.4.3.4, 3.4.6.4, 4.8^2, 3.12^2, and 3^4.6 uniform tilings, as well as the snub-632 and bew tilings. In several cases the results provide proofs for previously conjectured formulas.

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arXiv2018

The On-Line Encyclopedia of Integer Sequences

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.

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arXiv2017

Analysis of the gift exchange problem

In the gift exchange game there are n players and n wrapped gifts. When a player's number is called, that person can either choose one of the remaining wrapped gifts, or can "steal" a gift from someone who has already unwrapped it, subject to the restriction that no gift can be stolen more than a total of sigma times. The problem is to determine the number of ways that the game can be played out, for given values of sigma and n. Formulas and asymptotic expansions are given for these numbers. This work was inspired in part by a 2005 remark by Robert A. Proctor in the On-Line Encyclopedia of Integer Sequences.

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arXiv2016

Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences

Pisot sequences (sequences $a_n$ with initial terms $a_0=x, a_1=y$, and defined for $n>1$ by $a_n= \lfloor a_{n-1}^2/a_{n-2} + \frac{1}{2} \rfloor$) often satisfy linear recurrences with constant coefficients that are valid for all $n \geq 0$, but there are also cautionary examples where there is a linear recurrence that is valid for an initial range of values of $n$ but fails to be satisfied beyond that point, providing further illustrations of Richard Guy's celebrated "Strong Law of Small Numbers". In this paper we present a decision algorithm, fully implemented in an accompanying Maple program ({\tt Pisot.txt}), that first searches for a putative linear recurrence and then decides whether or not it holds for all values of $n$. We also explain why the failures happen (in some cases the `fake' linear recurrence may be valid for thousands of terms). We conclude by defining, and studying, higher-order analogs of Pisot sequences, and point out that similar phenomena occur there, albeit far less frequently. This article is dedicated to Richard K. Guy (b. Sept. 30, 1916) on his 100th birthday.

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arXiv2015

Odd-Rule Cellular Automata on the Square Grid

An "odd-rule" cellular automaton (CA) is defined by specifying a neighborhood for each cell, with the rule that a cell turns ON if it is in the neighborhood of an odd number of ON cells at the previous generation, and otherwise turns OFF. We classify all the odd-rule CAs defined by neighborhoods which are subsets of a 3 X 3 grid of square cells. There are 86 different CAs modulo trivial symmetries. When we consider only the different sequences giving the number of ON cells after n generations, the number drops to 48, two of which are the Moore and von Neumann CAs. This classification is carried out by using the "meta-algorithm" described in an earlier paper to derive the generating functions for the 86 sequences, and then removing duplicates. The fastest-growing of these CAs is neither the Fredkin nor von Neumann neighborhood, but instead is one defined by "Odd-rule" 365, which turns ON almost 75% of all possible cells.

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arXiv2015

The Yellowstone Permutation

Define a sequence of positive integers by the rule that a(n) = n for 1 <= n <= 3, and for n >= 4, a(n) is the smallest number not already in the sequence which has a common factor with a(n-2) and is relatively prime to a(n-1). We show that this is a permutation of the positive integers. The remarkable graph of this sequence consists of runs of alternating even and odd numbers, interrupted by small downward spikes followed by large upward spikes, suggesting the eruption of geysers in Yellowstone National Park. On a larger scale the points appear to lie on infinitely many distinct curves. There are several unanswered questions concerning the locations of these spikes and the equations for these curves.

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arXiv2015

On the Number of ON Cells in Cellular Automata

If a cellular automaton (CA) is started with a single ON cell, how many cells will be ON after n generations? For certain "odd-rule" CAs, including Rule 150, Rule 614, and Fredkin's Replicator, the answer can be found by using the combination of a new transformation of sequences, the run length transform, and some delicate scissor cuts. Several other CAs are also discussed, although the analysis becomes more difficult as the patterns become more intricate.

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