Encyclopedia Of Integer Sequences Research Articles

THE SUMMED PAPERFOLDING SEQUENCE

AbstractThe sequence $a( 1) ,a( 2) ,a( 3) ,\ldots, $ labelled A088431 in the Online Encyclopedia of Integer Sequences, is defined by: $a( n) $ is half of the $( n+1) $ th component, that is, the $( n+2) $ th term, of the continued fraction expansion of $$ \begin{align*} \sum_{k=0}^{\infty }\frac{1}{2^{2^{k}}}. \end{align*} $$ Dimitri Hendriks has suggested that it is the sequence of run lengths of the paperfolding sequence, A014577. This paper proves several results for this summed paperfolding sequence and confirms Hendriks’s conjecture.

Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries

Abstract Let un = un (k) denote the generalized Leonardo number defined recursively by un = un− 1 + un− 2 + k for n ≥ 2, where u 0 = u 1 = 1. Terms of the sequence un (1) are referred to simply as Leonardo numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1 ≤ k ≤ 7, where several connections are made to entries in the On-Line Encyclopedia of Integer Sequences. By Trudi’s formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case k = 1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures.

On a sequence of rational numbers with unusual divisibility by a power of 2

In this note we consider the sequence of rational numbers bn = ∑k=1n2k∕k. We show that the power of 2 in the expansion of bn is unusually large, at least n + 1 − log2(n + 1), and that this bound is best possible. The sequence bn, n = 1,2,3,…, is related to the sequence A0031449 in the On-Line Encyclopedia of Integer Sequences.

Exploring Polygonal Number Sieves through Computational Triangulation

The paper deals with the exploration of subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences. Eliminations are based on special rules similarly to how the classic sieve of Eratosthenes was developed through the elimination of multiples of primes. These elementary number theory activities, appropriate for technology-enhanced secondary mathematics education courses, are supported by a spreadsheet, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences. General formulas for subsequences of polygonal numbers referred to in the paper as polygonal number sieves of order k, that include base-two exponential functions of k, have been developed. Different problem-solving approaches to the derivation of such and other sieves based on the technology-immune/technology-enabled framework have been used. The accuracy of computations and mathematical reasoning is confirmed through the technique of computational triangulation enabled by using more than one digital tool. A few relevant excerpts from the history of mathematics are briefly featured.

Skew Dyck paths without up–down–left

Skew Dyck paths without up–down–left are enumerated. In a second step, the number of contiguous subwords ‘up–down–left’ are counted. This explains and extends results that were posted in the Encyclopedia of Integer Sequences.

Learning Program Synthesis for Integer Sequences from Scratch

We present a self-learning approach for synthesizing programs from integer sequences. Our method relies on a tree search guided by a learned policy. Our system is tested on the On-Line Encyclopedia of Integer Sequences. There, it discovers, on its own, solutions for 27987 sequences starting from basic operators and without human-written training examples.

Continued fractions and the Thomson problem

We introduce new analytical approximations of the minimum electrostatic energy configuration of n electrons, E(n), when they are constrained to be on the surface of a unit sphere. Using 453 putative optimal configurations, we searched for approximations of the form E(n) = (n^2/2) , e^{g(n)} where g(n) was obtained via a memetic algorithm that searched for truncated analytic continued fractions finally obtaining one with Mean Squared Error equal to {5.5549 times 10^{-8}} for the model of the normalized energy (E_n(n) equiv e^{g(n)} equiv 2E(n)/n^2). Using the Online Encyclopedia of Integer Sequences, we searched over 350,000 sequences and, for small values of n, we identified a strong correlation of the highest residual of our best approximations with the sequence of integers n defined by the condition that n^2+12 is a prime. We also observed an interesting correlation with the behavior of the smallest angle alpha (n), measured in radians, subtended by the vectors associated with the nearest pair of electrons in the optimal configuration. When using both sqrt{n} and alpha (n) as variables a very simple approximation formula for E_n(n) was obtained with MSE= 7.9963 times 10^{-8} and MSE= 73.2349 for E(n). When expanded as a power series in infinity, we observe that an unknown constant of an expansion as a function of n^{-1/2} of E(n) first proposed by Glasser and Every in 1992 as -1.1039, and later refined by Morris, Deaven and Ho as -1.104616 in 1996, may actually be very close to −1.10462553440167 when the assumed optima for nle 200 are used.

The integration of OEIS links in zbMATH Open

The transition towards an Open Data Platform enabled zbMATH Open to build a network of open resources. Important components in the evolving information system are mathematical research data, which are of quite heterogeneous nature. For their interlinking, zbMATH Open provides Application Programming Interface (API) solutions to offer mathematical research data to the community. Among other APIs recently implemented at zbMATH Open, the so-called Links API is aimed to document interconnections between our database and external platforms which display mathematical literature indexed at zbMATH Open. The Digital Library of Mathematical Functions (DLMF) has been our first partner and their data have been integrated in our database in 2021. Recently we interlinked with the second platform, the Online Encyclopedia of Integer Sequences (OEIS), a renowned digital database of number sequences that cites a lot of mathematical literature, especially from number theory and graph theory. The purpose of this short contribution is to announce and discuss the links to OEIS data in zbMATH Open.

The number of {1243, 2134}-avoiding permutations

We show that the counting sequence for permutations avoiding both of the (classical) patterns 1243 and 2134 has the algebraic generating function supplied by Vaclav Kotesovec for sequence A164651 in The On-Line Encyclopedia of Integer Sequences.

Chaos Game Representation

The chaos game representation (CGR) is an interesting method to visualize one-dimensional sequences. In this paper, we show how to construct a chaos game representation. The applications mentioned here are biological, in which CGR was able to uncover patterns in DNA or proteins that were previously unknown. We also show how CGR might be introduced in the classroom, either in a modelling course or in a dynamical systems course. Some sequences that are tested are taken from the Online Encyclopedia of Integer Sequences, and others are taken from sequences that arose mainly from a course in experimental mathematics.

Sets of Non-Self-Intersecting Paths Connecting the Vertices of a Convex Polygon

The paper deals with counting the sets of non-self-intersecting paths whose nodes form a partitioning of the set of vertices of a given convex polygon. There turn to exist compact formulae when the magnitude of these sets is fixed. Some of these formulae provide new properties for some of the entries of the On-line Encyclopedia of Integer Sequences, while others generate new entries therein.

On digital sequences associated with Pascal’s triangle

We consider the sequence of integers whose nth term has base-p expansion given by the nth 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 involving alternating sum-of-digits modulo p. This note ends with a discussion about Pascal’s pyramid built with trinomial coefficients.

Combinatorial aspects of sandpile models on wheel and fan graphs

We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations of the model’s recurrent configurations on these families. For wheel graphs, we exhibit a bijection between these recurrent configurations and the set of subgraphs of the cycle graph which maps the level of the configuration to the number of edges of the subgraph. This bijection relies on two key ingredients. The first consists in considering a stochastic variant of the standard Abelian sandpile model (ASM), rather than the ASM itself. The second ingredient is a mapping from a given recurrent state to a canonical minimal recurrent state, exploiting similar ideas to previous studies of the ASM on complete bipartite graphs and Ferrers graphs. We also show that on the wheel graph with 2n vertices, the number of recurrent states with level n is given by the first differences of the central Delannoy numbers. Finally, using similar tools, we exhibit a bijection between the set of recurrent configurations of the ASM on fan graphs and the set of subgraphs of the path graph containing the right-most vertex of the path. We show that these sets are also equinumerous with certain lattice paths, which we name Kimberling paths after the author of the corresponding entry in the Online Encyclopedia of Integer Sequences.

Wiener Index of Some Brooms

In the field of chemical graph theory, a Wiener (topological) index is a type of a molecular descriptor that is calculated based on the molecular graph of alkanes. It gives the sum of geodesic distances (or shortest paths) between all pairs of vertices of the graph. We found and prove the Wiener indices of some Brooms, which are Caterpillars, giving several unknown sequences that are now added to the collection of the largest Online Encyclopedia of Integer Sequences.

On some combinatorial sequences associated to invariant theory

We study the enumerative and analytic properties of some sequences constructed using tensor invariant theory. The octant sequences are constructed from the exceptional Lie group G2 and the quadrant sequences from the special linear group SL(3). In each case we show that the corresponding sequences are related by binomial transforms. The first three octant sequences and the first four quadrant sequences are listed in the On-Line Encyclopedia of Integer Sequences (OEIS). These sequences all have interpretations as enumerating two-dimensional lattice walks but for the octant sequences the boundary conditions are unconventional. These sequences are all P-recursive and we give the corresponding recurrence relations. In all cases the associated differential operators are of third order and have the remarkable property that they can be solved to give closed formulae for the ordinary generating functions in terms of classical Gaussian hypergeometric functions. Moreover, we show that the octant sequences and the quadrant sequences are related by the branching rules for the inclusion of SL(3) in G2.

Classifying Groups With a Small Number of Subgroups

We provide lower bounds on the number of subgroups of a group G as a function of the primes and exponents appearing in the prime factorization of . Using these bounds, we classify all abelian groups with 22 or fewer subgroups, and all nonabelian groups with 19 or fewer subgroups. This allows us to extend the integer sequence A274847 in the On-Line Encyclopedia of Integer Sequences introduced by Slattery.

Zavadskij modules over cluster-tilted algebras of type $ \mathbb{A} $

<abstract><p>Zavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra $ A $. In this setting, we prove that the indecomposable uniserial $ A $-modules with a mast of multiplicity one in each vertex are Zavadskij modules. Since the Zavadskij property carries over to direct summands, but it is not invariant under the formation of direct sums, we give a criterion to determine when the direct sum of indecomposable Zavadskij modules is again a Zavadskij module. In addition, we use the triangulations of the $ n+3 $-gon associated with the cluster-tilted algebra of type $ \mathbb{A}_{n} $ to give a formula for the number of indecomposable Zavadskij modules over any cluster-tilted algebra of type $ \mathbb{A}_{n} $. In this case, the formula gives the dimension of the cluster-tilted algebra. As an application, we discuss some integer sequences in the OEIS (The On-Line Encyclopedia of Integer Sequences) that allow us to enumerate indecomposable Zavadskij modules.</p></abstract>

ALMOST SQUARING THE SQUARE: OPTIMAL PACKINGS FOR NON-DECOMPOSABLE SQUARES

We consider the problem of finding the minimum uncovered area (trim loss) when tiling non- overlapping distinct integer-sided squares in an N × N square container such that the squares are placed with their edges parallel to those of the container. We find such trim losses and associated optimal packings for all container sizes N from 1 to 101, through an independently developed adaptation of Ian Gambini’s enumerative algorithm. The results were published as a new sequence to The On-Line Encyclopedia of Integer Sequences®. These are the first known results for optimal packings in non-decomposable squares.

A note on the Neyman–Rayner triangle

This note raises questions for other number theorists to tackle. It considers a triangle arising from some statistical research of John Rayner and his use of some orthonormal polynomials related to the Legendre polynomials. These are expressed in a way that challenges the generalizing them. In particular, the coefficients are expressed in a triangle and related to known sequences in the Online Encyclopedia of Integer Sequences. The note actually raises more questions than it answers when it links with the cluster algebra of Fomin and Zelevinsky.

Using Technology With Future Teachers Of Primary School Mathematics: A Glimpse Into An American Experience

The paper describes the author’s experience in using technology within mathematics content and methods (undergraduate and graduate) courses for prospective teachers of primary grades (age 5–10). The main pedagogical idea behind the courses is to change pre-teachers’ perception of mathematics as a subject matter most people predictably dislike. It is suggested that technology can assist instructors in making mathematics an enjoyable subject matter without sacrificing content. The paper provides examples of using Excel, Wolfram Alpha, dynamic geometry software, computer graphing program, and the Online Encyclopedia of Integer Sequences. In conclusion, solicited comments by teacher candidates about their experience of learning computer assisted mathematics are shared.