scholarly journals A note on the Neyman–Rayner triangle

2021 ◽  
Vol 27 (4) ◽  
pp. 164-166
Author(s):  
A. G. Shannon ◽  
Keyword(s):  
Legendre Polynomials ◽  
Cluster Algebra ◽  
Integer Sequences ◽  
Statistical Research ◽  
Orthonormal Polynomials ◽  
Online Encyclopedia

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.

scholarly journals Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

10.37236/1052 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Brad Jackson ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Binary Trees ◽  
Integer Sequences ◽  
Number Of Leaves ◽  
Online Encyclopedia ◽  
Fibonacci Sequences ◽  
Infinite Sequences

We consider 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).

scholarly journals Polygonal Numbers

2013 ◽  
Vol 21 (2) ◽  
pp. 103-113 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Related Result ◽  
Integer Sequences ◽  
Triangular Numbers ◽  
Perfect Numbers ◽  
Polygonal Numbers ◽  
Online Encyclopedia ◽  
Four Square ◽  
Mersenne Primes ◽  
Ordinal Index

Summary In the article the formal characterization of triangular numbers (famous from [15] and words “EYPHKA! num = Δ+Δ+Δ”) [17] is given. Our primary aim was to formalize one of the items (#42) from Wiedijk’s Top 100 Mathematical Theorems list [33], namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar language evolved and attributes with arguments were implemented, we decided to extend these lines and we characterized polygonal numbers. We formalized centered polygonal numbers, the connection between triangular and square numbers, and also some equalities involving Mersenne primes and perfect numbers. We gave also explicit formula to obtain from the polygonal number its ordinal index. Also selected congruences modulo 10 were enumerated. Our work basically covers the Wikipedia item for triangular numbers and the Online Encyclopedia of Integer Sequences (http://oeis.org/A000217). An interesting related result [16] could be the proof of Lagrange’s four-square theorem or Fermat’s polygonal number theorem [32].

scholarly journals On some results concerning the polygonal polynomials

2019 ◽  
Vol 35 (1) ◽  
pp. 01-12
Author(s):  
DORIN ANDRICA ◽  
◽  
OVIDIU BAGDASAR ◽  
Keyword(s):  
Recurrence Relations ◽  
Integer Sequences ◽  
Open Questions ◽  
New Meanings ◽  
Online Encyclopedia ◽  
Integral Formulae

In this paper we define the nth polygonal polynomial and we investigate recurrence relations and exact integral formulae for the coefficients of Pn and for those of the Mahonianpolynomials. We also explore numerical properties of these coefficients, unraveling new meanings for old sequences and generating novel entries to the Online Encyclopedia of Integer Sequences (OEIS). Some open questions are also formulated.

scholarly journals A New Record of Graph Enumeration Enabled by Parallel Processing

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1214 ◽  
Author(s):  
Zhipeng Xu ◽  
Xiaolong Huang ◽  
Fabian Jimenez ◽  
Yuefan Deng
Keyword(s):  
Parallel Processing ◽  
Shortest Path ◽  
Interconnection Networks ◽  
New Record ◽  
Parallel Computers ◽  
Regular Graphs ◽  
Integer Sequences ◽  
Path Lengths ◽  
Online Encyclopedia ◽  
Processor Cores

Using three supercomputers, we broke a record set in 2011, in the enumeration of non-isomorphic regular graphs by expanding the sequence of A006820 in the Online Encyclopedia of Integer Sequences (OEIS), to achieve the number for 4-regular graphs of order 23 as 429,668,180,677,439, while discovering several regular graphs with minimum average shortest path lengths (ASPL) that can be used as interconnection networks for parallel computers. The enumeration of 4-regular graphs and the discovery of minimal-ASPL graphs are extremely time consuming. We accomplish them by adapting GENREG, a classical regular graph generator, to three supercomputers with thousands of processor cores.

scholarly journals On Partially Ordered Patterns of Length 4 and 5 in Permutations

10.37236/8605 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Alice L. L. Gao ◽  
Sergey Kitaev
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Relative Order ◽  
Permutation Patterns ◽  
Integer Sequences ◽  
Arbitrary Length ◽  
Partially Ordered ◽  
Long Line ◽  
Online Encyclopedia ◽  
Wilf Equivalence

Partially ordered patterns (POPs) generalize the notion of classical patterns studied widely in the literature in the context of permutations, words, compositions and partitions. In an occurrence of a POP, the relative order of some of the elements is not important. Thus, any POP of length $k$ is defined by a partially ordered set on $k$ elements, and classical patterns correspond to $k$-element chains. The notion of a POP provides  a convenient language to deal with larger sets of permutation patterns. This paper contributes to a long line of research on classical permutation patterns of length 4 and 5, and beyond, by conducting a systematic search of connections between sequences in the Online Encyclopedia of Integer Sequences (OEIS) and permutations avoiding POPs of length 4 and 5. As the result, we (i) obtain  13 new enumerative results for classical patterns of length 4 and 5, and a number of results for patterns of arbitrary length, (ii) collect under one roof many sporadic results in the literature related to avoidance of patterns of length 4 and 5, and (iii) conjecture 6 connections to the OEIS. Among the most intriguing bijective questions we state, 7 are related to explaining Wilf-equivalence of various sets of patterns, e.g. 5 or 8 patterns of length 4, and 2 or 6 patterns of length 5.

Application of Legendre Polynomials to the Elimination of Baseline Variation in Biological FT-IR Spectra

1988 ◽  
Vol 42 (3) ◽  
pp. 395-400 ◽  
Author(s):  
Alan H. Lipkus
Keyword(s):  
Ir Spectra ◽  
Legendre Polynomials ◽  
Orthonormal Polynomials ◽  
Low Degree ◽  
Ft Ir ◽  
Protein Spectra ◽  
Low Degree Polynomials

A method for eliminating baseline variation is proposed for use on biological FT-IR spectra. This method is motivated by the fact that baselines are often assumed to be low-degree polynomials. Spectra are expanded in terms of a set of orthonormal polynomials derived from the Legendre polynomials, and leading terms of the expansion, which contain most of the baseline variation, are removed. An application of this method to protein spectra is presented and discussed.

scholarly journals On Fractional Orthonormal Polynomials of a Discrete Variable

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
I. Area ◽  
J. D. Djida ◽  
J. Losada ◽  
Juan J. Nieto
Keyword(s):  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Discrete Variable ◽  
Orthonormal Polynomials ◽  
Basic Properties

A fractional analogue of classical Gram or discrete Chebyshev polynomials is introduced. Basic properties as well as their relation with the fractional analogue of Legendre polynomials are presented.

Asymptotic behavior of integer sequences related to knots and (m,n)-anacci constants

2018 ◽  
Vol 27 (07) ◽  
pp. 1841011
Author(s):  
Igor Szczyrba
Keyword(s):  
Asymptotic Behavior ◽  
Affine Geometry ◽  
Integer Sequences ◽  
Linear Recurrences ◽  
Online Encyclopedia

We study the asymptotic behavior of integer sequences related to knots that are generated by linear recurrences. We determine which of the 85 such sequences cataloged in Online Encyclopedia of Integer Sequences have ratios of their consecutive terms converging to a limit. We show that all but one of the ratio limits can be expressed by means of the [Formula: see text]-anacci constants with [Formula: see text] equal to 1 or 2. Finally, we demonstrate how the [Formula: see text]-anacci constants are linked to affine geometry.

Remainder sum and quotient sum function

2015 ◽  
Vol 07 (01) ◽  
pp. 1550001
Author(s):  
A. David Christopher
Keyword(s):  
Positive Integer ◽  
Periodic Function ◽  
Integer Partition ◽  
Integer Sequences ◽  
Partition Theory ◽  
Arithmetical Functions ◽  
Finite Set ◽  
Online Encyclopedia ◽  
Divisibility Properties ◽  
Positive Integers

This paper is concerned with two arithmetical functions namely remainder sum function and quotient sum function which are respectively the sequences A004125 and A006218 in Online Encyclopedia of Integer Sequences. The remainder sum function is defined by [Formula: see text] for every positive integer n, and quotient sum function is defined by [Formula: see text] where q(n, i) is the quotient obtained when n is divided by i. We establish few divisibility properties these functions enjoy and we found their bounds. Furthermore, we define restricted remainder sum function by RA(n) = ∑k∈A n mod k where A is a set of positive integers and we define restricted quotient sum function by QA(n) = ∑k∈A q(n, k). The function QA(n) is found to be a quasi-polynomial of degree one when A is a finite set of positive integers and RA(n) is found to be a periodic function with period ∏a∈A a. Finally, the above defined four functions found to have recurrence relation whose derivation requires few results from integer partition theory.

scholarly journals On Generalized Lucas Pseudoprimality of Level k

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 838
Author(s):  
Dorin Andrica ◽  
Ovidiu Bagdasar
Keyword(s):  
Integer Sequences ◽  
Basic Properties ◽  
Arithmetic Properties ◽  
Lucas Sequences ◽  
Online Encyclopedia ◽  
The Relationship ◽  
Different Levels ◽  
Lucas Pseudoprimes

We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some recent arithmetic properties of the generalized Lucas, and generalized Pell–Lucas sequences, to define some new types of pseudoprimes of levels k+ and k− and parameter a. For these novel pseudoprime sequences we investigate some basic properties and calculate numerous associated integer sequences which we have added to the Online Encyclopedia of Integer Sequences.

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