Figurate Numbers Within the Light Superscripts
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Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube. -
An Heptagonal Numbers
© April 2021| IJIRT | Volume 7 Issue 11 | ISSN: 2349-6002 An Heptagonal Numbers Dr. S. Usha Assistant Professor in Mathematics, Bon Secours Arts and Science College for Women, Mannargudi, Affiliated by Bharathidasan University, Tiruchirappalli, Tamil Nadu Abstract - Two Results of interest (i)There exists an prime factor that divides n is one. First few square free infinite pairs of heptagonal numbers (푯풎 , 푯풌) such that number are 1,2,3,5,6,7,10,11,13,14,15,17…… their ratio is equal to a non –zero square-free integer and (ii)the general form of the rank of square heptagonal Definition(1.5): ퟑ ퟐ풓+ퟏ number (푯 ) is given by m= [(ퟏퟗ + ퟑ√ퟒퟎ) + 풎 ퟐퟎ Square free integer: ퟐ풓+ퟏ (ퟏퟗ − ퟑ√ퟒퟎ) +2], where r = 0,1,2…….relating to A Square – free Integer is an integer which is divisible heptagonal number are presented. A Few Relations by no perfect Square other than 1. That is, its prime among heptagonal and triangular number are given. factorization has exactly one factors for each prime that appears in it. For example Index Terms - Infinite pairs of heptagonal number, the 10 =2.5 is square free, rank of square heptagonal numbers, square-free integer. But 18=2.3.3 is not because 18 is divisible by 9=32 The smallest positive square free numbers are I. PRELIMINARIES 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23……. Definition(1.1): A number is a count or measurement heptagon: Definition(1.6): A heptagon is a seven –sided polygon. -
Elementary School Numbers and Numeration
DOCUMENT RESUME ED 166 042 SE 026 555 TITLE Mathematics for Georgia Schools,' Volume II: Upper Elementary 'Grades. INSTITUTION Georgia State Dept. of Education, Atlanta. Office of Instructional Services. e, PUB DATE 78 NOTE 183p.; For related document, see .SE 026 554 EDRB, PRICE MF-$0.33 HC-$10.03 Plus Postage., DESCRIPTORS *Curriculm; Elementary Educatia; *Elementary School Mathematics; Geometry; *Instruction; Meadurement; Number Concepts; Probability; Problem Solving; Set Thory; Statistics; *Teaching. Guides ABSTRACT1 ' This guide is organized around six concepts: sets, numbers and numeration; operations, their properties and number theory; relations and functions; geometry; measurement; and probability and statistics. Objectives and sample activities are presented for.each concept. Separate sections deal with the processes of problem solving and computation. A section on updating curriculum includes discussion of continuing program improvement, evaluation of pupil progress, and utilization of media. (MP) ti #######*#####*########.#*###*######*****######*########*##########**#### * Reproductions supplied by EDRS are the best that can be made * * from the original document. * *********************************************************************** U S DEPARTMENT OF HEALTH, EDUCATION & WELFARE IS NATIONAL INSTITUTE OF.. EDUCATION THIS DOCUMENT HA4BEEN REPRO- DuCED EXACTLY AS- RECEIVEDFROM THE PERSON OR ORGANIZATIONORIGIN- ATING IT POINTS OF VIEWOR 01NIONS STATED DO NOT NECESSARILYEpRE SENT OFFICIAL NATIONAL INSTITUTEOF TO THE EDUCATION4 -
Figurate Numbers
Figurate Numbers by George Jelliss June 2008 with additions November 2008 Visualisation of Numbers The visual representation of the number of elements in a set by an array of small counters or other standard tally marks is still seen in the symbols on dominoes or playing cards, and in Roman numerals. The word "calculus" originally meant a small pebble used to calculate. Bear with me while we begin with a few elementary observations. Any number, n greater than 1, can be represented by a linear arrangement of n counters. The cases of 1 or 0 counters can be regarded as trivial or degenerate linear arrangements. The counters that make up a number m can alternatively be grouped in pairs instead of ones, and we find there are two cases, m = 2.n or 2.n + 1 (where the dot denotes multiplication). Numbers of these two forms are of course known as even and odd respectively. An even number is the sum of two equal numbers, n+n = 2.n. An odd number is the sum of two successive numbers 2.n + 1 = n + (n+1). The even and odd numbers alternate. Figure 1. Representation of numbers by rows of counters, and of even and odd numbers by various, mainly symmetric, formations. The right-angled (L-shaped) formation of the odd numbers is known as a gnomon. These do not of course exhaust the possibilities. 1 2 3 4 5 6 7 8 9 n 2 4 6 8 10 12 14 2.n 1 3 5 7 9 11 13 15 2.n + 1 Triples, Quadruples and Other Forms Generalising the divison into even and odd numbers, the counters making up a number can of course also be grouped in threes or fours or indeed any nonzero number k. -
TRIANGULAR REPUNIT—THERE IS but 1 a Repunit Is Any Integer That
Czechoslovak Mathematical Journal, 60 (135) (2010), 1075–1077 TRIANGULAR REPUNIT—THERE IS BUT 1 John H. Jaroma, Ave Maria (Received June 27, 2009) Abstract. In this paper, we demonstrate that 1 is the only integer that is both triangular and a repunit. Keywords: repunit, triangular number MSC 2010 : 11A99 A repunit is any integer that can be written in decimal notation as a string of 1’s. Examples include 1, 11, 111, 1111, 11111,... Furthermore, if we denote rn to be the nth repunit then it follows that 10n − 1 (1) rn = . 9 Triangular numbers are those integers that can be represented by the number of dots evenly arranged in an equilateral triangle. More specifically, they are the numbers 1, 3, 6, 10, 15,... It follows that the nth triangular number, tn is the sum of the first n consecutive integers beginning with 1. In particular, for n > 1 n(n + 1) (2) tn = . 2 Now, the number 1 is both triangular and a repunit. It is the objective of this note to illustrate that 1 is the only such number. To this end, we state and prove the following lemma. It incorporates a result that [1] asserts had been known by the early Pythagoreans. Appearing in Platonic Questions, Plutarch states that eight times any triangular number plus one is a square. The result is actually necessary and sufficient. We demonstrate it as such here. 1075 Lemma. The integer n is triangular if and only if 8n +1 is a square. Proof. If n = tn, then by (2), n(n + 1) 8 +1=4n2 +4n +1=(2n + 1)2. -
Hierarchical Self-Assembly of Telechelic Star Polymers: from Soft Patchy Particles to Gels and Diamond Crystals
Home Search Collections Journals About Contact us My IOPscience Hierarchical self-assembly of telechelic star polymers: from soft patchy particles to gels and diamond crystals This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 New J. Phys. 15 095002 (http://iopscience.iop.org/1367-2630/15/9/095002) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 131.130.87.134 This content was downloaded on 18/01/2017 at 13:41 Please note that terms and conditions apply. You may also be interested in: A systematic coarse-graining strategy for semi-dilute copolymer solutions: from monomersto micelles Barbara Capone, Ivan Coluzza and Jean-Pierre Hansen Polymer solutions: from hard monomers to soft polymers Jean-Pierre Hansen, Chris I Addison and Ard A Louis Computer simulations of supercooled polymer melts in the bulk and in confinedgeometry J Baschnagel and F Varnik Theoretical models for bridging timescales in polymer dynamics M G Guenza Structural properties of dendrimer–colloid mixtures Dominic A Lenz, Ronald Blaak and Christos N Likos Correlations in concentrated dendrimer solutions I O Götze and C N Likos Topological characteristics of model gels Mark A Miller, Ronald Blaak and Jean-Pierre Hansen Computer simulations of polyelectrolyte stars and brushes Christos N Likos, Ronald Blaak and Aaron Wynveen Conformational and dynamical properties of ultra-soft colloids in semi-dilute solutions under shear flow Sunil P Singh, Dmitry A Fedosov, Apratim -
~Umbers the BOO K O F Umbers
TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex planation of the multitudinous ways in which the word "number" is used. -
PELL FACTORIANGULAR NUMBERS Florian Luca, Japhet Odjoumani
PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 105(119) (2019), 93–100 DOI: https://doi.org/10.2298/PIM1919093L PELL FACTORIANGULAR NUMBERS Florian Luca, Japhet Odjoumani, and Alain Togbé Abstract. We show that the only Pell numbers which are factoriangular are 2, 5 and 12. 1. Introduction Recall that the Pell numbers Pm m> are given by { } 1 αm βm P = − , for all m > 1, m α β − where α =1+ √2 and β =1 √2. The first few Pell numbers are − 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025,... Castillo [3], dubbed a number of the form Ftn = n!+ n(n + 1)/2 for n > 1 a factoriangular number. The first few factoriangular numbers are 2, 5, 12, 34, 135, 741, 5068, 40356, 362925,... Luca and Gómez-Ruiz [8], proved that the only Fibonacci factoriangular numbers are 2, 5 and 34. This settled a conjecture of Castillo from [3]. In this paper, we prove the following related result. Theorem 1.1. The only Pell numbers which are factoriangular are 2, 5 and 12. Our method is similar to the one from [8]. Assuming Pm = Ftn for positive integers m and n, we use a linear forms in p-adic logarithms to find some bounds on m and n. The resulting bounds are large so we run a calculation to reduce the bounds. This computation is highly nontrivial and relates on reducing the diophantine equation Pm = Ftn modulo the primes from a carefully selected finite set of suitable primes. 2010 Mathematics Subject Classification: Primary 11D59, 11D09, 11D7; Secondary 11A55. -
Patterns in Figurate Sequences
Patterns in Figurate Sequences Concepts • Numerical patterns • Figurate numbers: triangular, square, pentagonal, hexagonal, heptagonal, octagonal, etc. • Closed form representation of a number sequence • Function notation and graphing • Discrete and continuous data Materials • Chips, two-color counters, or other manipulatives for modeling patterns • Student activity sheet “Patterns in Figurate Sequences” • TI-73 EXPLORER or TI-83 Plus/SE Introduction Mathematics has been described as the “science of patterns.” Patterns are everywhere and may appear as geometric patterns or numeric patterns or both. Figurate numbers are examples of patterns that are both geometric and numeric since they relate geometric shapes of polygons to numerical patterns. In this activity you will analyze, extend, and describe patterns involving figurate numbers and make connections between numeric and geometric representations of patterns. PTE: Algebra Page 1 © 2003 Teachers Teaching With Technology Patterns in Figurate Sequences Student Activity Sheet 1. Using chips or other manipulatives, analyze the following pattern and extend the pattern pictorially for two more terms. • • • • • • • • • • 2. Write the sequence of numbers that describes the quantity of dots above. 3. Describe this pattern in another way. 4. Extend and describe the following pattern with pictures, words, and numbers. • • • • • • • • • • • • • • 5. Analyze Table 1. Fill in each of the rows of the table. Table 1: Figurate Numbers Figurate 1st 2nd 3rd 4th 5th 6th 7th 8th nth Number Triangular 1 3 6 10 15 21 28 36 n(n+1)/2 Square 1 4 9 16 25 36 49 64 Pentagonal 1 5 12 22 35 51 70 Hexagonal 1 6 15 28 45 66 Heptagonal 1 7 18 34 55 Octagonal 1 8 21 40 Nonagonal 1 9 24 Decagonal 1 10 Undecagonal 1 .. -
Identifying Figurate Number Patterns
Identifying Figurate Lesson 7-9 Number Patterns DATE TIME SRB 1 Draw the next three rectangular numbers. 58-61 2 a. Complete the list of the first 10 rectangular numbers. 2, 6, 12, 20, 30, 42 56,EM4_MJ2_G4_U07_L09_001A.ai 72, 90, 110 b. How did you get your answers? Sample answer: I kept multiplying the next 2 counting numbers. I knew 42 = 6 ⁎ 7, so to find the next rectangular number, I multiplied 7 ⁎ 8 = 56. 3 a. Continue the following pattern: 2 = 2 2 + 4 = 6 2 + 4 + 6 = 12 2 + 4 + 6 + 8 = 20 2 + 4 + 6 + 8 + 10 = 30 2 + 4 + 6 + 8 + 10 + 12 = 42 b. Describe the pattern. Sample answer: Each rectangular number is the sum of the even numbers in order, starting with 2. c. What pattern do you notice about the number of addends and the rectangular number? The number of addends is the same as the “number” of the rectangular number. For example, 30 is the fifth rectangular number, and it has 5 addends. d. Describe a similar pattern with square numbers. Sample answer: Each square number is the sum of the odd numbers in order, starting with 1. 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16, and so on. 252 4.OA.5, 4.NBT.6, SMP7, SMP8 Identifying Figurate Lesson 7-9 DATE TIME Number Patterns (continued) Try This 4 Triangular numbers are numbers that are the sum of consecutive counting numbers. For example, the triangular number 3 is the sum of 1 + 2, and the triangular number 6 is the sum of 1 + 2 + 3. -
Generalized Sum of Stella Octangula Numbers
DOI: 10.5281/zenodo.4662348 Generalized Sum of Stella Octangula Numbers Amelia Carolina Sparavigna Department of Applied Science and Technology, Politecnico di Torino A generalized sum is an operation that combines two elements to obtain another element, generalizing the ordinary addition. Here we discuss that concerning the Stella Octangula Numbers. We will also show that the sequence of these numbers, OEIS A007588, is linked to sequences OEIS A033431, OEIS A002378 (oblong or pronic numbers) and OEIS A003154 (star numbers). The Cardano formula is also discussed. In fact, the sequence of the positive integers can be obtained by means of Cardano formula from the sequence of Stella Octangula numbers. Keywords: Groupoid Representations, Integer Sequences, Binary Operators, Generalized Sums, OEIS, On-Line Encyclopedia of Integer Sequences, Cardano formula. Torino, 5 April 2021. A generalized sum is a binary operation that combines two elements to obtain another element. In particular, this operation acts on a set in a manner that its two domains and its codomain are the same set. Some generalized sums have been previously proposed proposed for different sets of numbers (Fibonacci, Mersenne, Fermat, q-numbers, repunits and others). The approach was inspired by the generalized sums used for entropy [1,2]. The analyses of sequences of integers and q-numbers have been collected in [3]. Let us repeat here just one of these generalized sums, that concerning the Mersenne n numbers [4]. These numbers are given by: M n=2 −1 . The generalized sum is: M m⊕M n=M m+n=M m+M n+M m M n In particular: M n⊕M 1=M n +1=M n+M 1+ M n M 1 1 DOI: 10.5281/zenodo.4662348 The generalized sum is the binary operation which is using two Mersenne numbers to have another Mersenne number. -
Notations Used 1
NOTATIONS USED 1 NOTATIONS ⎡ (n −1)(m − 2)⎤ Tm,n = n 1+ - Gonal number of rank n with sides m . ⎣⎢ 2 ⎦⎥ n(n +1) T = - Triangular number of rank n . n 2 1 Pen = (3n2 − n) - Pentagonal number of rank n . n 2 2 Hexn = 2n − n - Hexagonal number of rank n . 1 Hep = (5n2 − 3n) - Heptagonal number of rank n . n 2 2 Octn = 3n − 2n - Octagonal number of rank n . 1 Nan = (7n2 − 5n) - Nanogonal number of rank n . n 2 2 Decn = 4n − 3n - Decagonal number of rank n . 1 HD = (9n 2 − 7n) - Hendecagonal number of rank n . n 2 1 2 DDn = (10n − 8n) - Dodecagonal number of rank n . 2 1 TD = (11n2 − 9n) - Tridecagonal number of rank n . n 2 1 TED = (12n 2 −10n) - Tetra decagonal number of rank n . n 2 1 PD = (13n2 −11n) - Pentadecagonal number of rank n . n 2 1 HXD = (14n2 −12n) - Hexadecagonal number of rank n . n 2 1 HPD = (15n2 −13n) - Heptadecagonal number of rank n . n 2 NOTATIONS USED 2 1 OD = (16n 2 −14n) - Octadecagonal number of rank n . n 2 1 ND = (17n 2 −15n) - Nonadecagonal number of rank n . n 2 1 IC = (18n 2 −16n) - Icosagonal number of rank n . n 2 1 ICH = (19n2 −17n) - Icosihenagonal number of rank n . n 2 1 ID = (20n 2 −18n) - Icosidigonal number of rank n . n 2 1 IT = (21n2 −19n) - Icositriogonal number of rank n . n 2 1 ICT = (22n2 − 20n) - Icositetragonal number of rank n . n 2 1 IP = (23n 2 − 21n) - Icosipentagonal number of rank n .