1.17 Triangle Numbers
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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. -
Induction for Tetrahedral Numbers
Induction for Tetrahedral Numbers Teacher Notes and Answers 7 8 9 10 11 12 TI-84PlusCE™ Investigation Student 50 min Teacher Notes: “Inductive Tetrahedrals” is part two of this three-part series. [Refer to Inductive Whole Numbers for Part One]. Part two follows a similar format to part one but is slightly more challenging from an algebraic perspective. Teachers can use part two in this series as a more independent environment for students focusing on providing feedback rather than instruction. A set of Power Point slides is provided that can be used to enhance the visual aspect of this lesson, the ability to visualise solutions can give problems more meaning. “Cubes and Squares” is part three of this three-part series. This section is designed as an assessment tool, marks have been allocated to each question. The questions are more reflective of the types of questions that students might experience in the short answer section of an examination where calculators may be used to derive answers or to simply verify by-hand solutions. A set of Power Point slides is provided that can be used to introduce students to this task. Introduction The purpose of this activity is to use exploration and observation to establish a rule for the sum of the first n tetrahedral numbers then use proof by mathematical induction to show that the rule is true for all whole numbers. What are the Tetrahedral numbers? The prefix ‘tetra’ refers to the quantity four, so it is not surprising that a tetrahedron consists of four faces, each face is a triangle. -
Input for Carnival of Math: Number 115, October 2014
Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144. -
Tetrahedral Numbers
Tetrahedral Numbers Student Worksheet TI-30XPlus Activity Student 50 min 7 8 9 10 11 12 MathPrint™ Finding Patterns What are the Tetrahedral numbers? The prefix ‘tetra’ refers to the quantity four, so it is not surprising that a tetrahedron consists of four faces, each face is a triangle. This triangular formation can sometimes be found in stacks of objects. The series of diagrams below shows the progression from one layer to the next for a stack of spheres. Row Number 1 2 3 4 5 Items Added Complete Stack Question: 1. Create a table of values for the row number and the corresponding quantity of items in a complete stack. Question: 2. Create a table of values for the row number and the corresponding quantity of items that are added to the stack. Question: 3. The calculator screen shown here illustrates how to determine the fifth tetrahedral number. The same command could be used to determine any of the tetrahedral numbers. Explain how this command is working. Question: 4. Verify that the calculation shown opposite is the same as the one generated in Question 3. Question: 5. Enter the numbers 1, 2 … 10 in List 1 on the calculator. Enter the first 10 tetrahedral numbers in List 2. Once the values have been entered try the following: a) Quadratic regression using List 1 and List 2. Check the validity of the result via substitution. b) Cubic regression using List 1 and List 2. Check the validity of the result via substitution. Texas Instruments 2021. You may copy, communicate and modify this material for non-commercial educational purposes Author: P. -
Newsletter 91
Newsletter 9 1: December 2010 Introduction This is the final nzmaths newsletter for 2010. It is also the 91 st we have produced for the website. You can have a look at some of the old newsletters on this page: http://nzmaths.co.nz/newsletter As you are no doubt aware, 91 is a very interesting and important number. A quick search on Wikipedia (http://en.wikipedia.org/wiki/91_%28number%29) will very quickly tell you that 91 is: • The atomic number of protactinium, an actinide. • The code for international direct dial phone calls to India • In cents of a U.S. dollar, the amount of money one has if one has one each of the coins of denominations less than a dollar (penny, nickel, dime, quarter and half dollar) • The ISBN Group Identifier for books published in Sweden. In more mathematically related trivia, 91 is: • the twenty-seventh distinct semiprime. • a triangular number and a hexagonal number, one of the few such numbers to also be a centered hexagonal number, and it is also a centered nonagonal number and a centered cube number. It is a square pyramidal number, being the sum of the squares of the first six integers. • the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): 91 = 6 3+(-5) 3 = 43+33. • the smallest positive integer expressible as a sum of six distinct squares: 91 = 1 2+2 2+3 2+4 2+5 2+6 2. -
On Ternary Cubic Equation
ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 9, March 2014 ON TERNARY CUBIC EQUATION P. Thirunavukarasu, S. Sriram Assistant Professor -P.G & Research Department of Mathematics, Periyar E.V.R College Tiruchirappalli – 620 023, Tamilnadu, India Assistant Professor–P.G & Research Department of Mathematics, National College, Tiruchirappalli – 620 001, Tamilnadu, India equations are analyzed for the non-trivial integral Abstract we obtain the non-trivial integral solutions for the solutions. These results have motivated us to search for ternary cubic equation non-trivial integral solutions of their varieties of ternary cubic Diophantine equation. This paper concerns with the problem of determining non-trivial integral solutions of . the equation with three unknowns given by A few interesting relations among the solutions are presented. Index Terms: Ternary Cubic, integral solutions, Pell’s explicit integral solutions of the above equation are form, nasty numbers presented. A few interesting relations among the solutions Notations are obtained. Oblong number of rank n obln n n 1 II. METHOD OF ANALYSIS Tetrahedral number of rank The ternary cubic equation under consideration is n n12 n n Tet n 6 Triangular number of rank (1) Taking (2) Polygonal number of rank n with sides nm12 (3) m tnmn, 1 2 We get Square pyramidal number of rank n n1 2 n 1 (4) Again taking the transformation n Sqpn 6 Pentagonal pyramidal number of rank Star number = and apply in (4) we get (5) 2 Stella Octangula number = St. oct n 2 n 1 n It is well known that the general form of the integral solutions of the Pellian equation. -
~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. -
The Frobenius Number for Sequences of Binomial Coefficients
The Frobenius number for sequences of binomial coefficients Aureliano M. Robles-Perez´ Universidad de Granada A talk based on a joint work with Jose´ Carlos Rosales (J. Number Theory 186 (2018) 473–492.) INdAM meeting: International meeting on numerical semigroups - Cortona 2018 3-7th September 2018 A. M. Robles-Perez´ (UGR) Frobenius number and binomial sequences IMNS-2018 1 / 29 Frobenius coin problem I Given relatively prime positive integers a1;:::;an, n ≥ 2, find a formula to compute the largest integer that is not representable as a non-negative integer linear combination of a1;:::;an. I F(a1;:::;an) (the Frobenius number of the set fa1;:::;ang) denotes the solution of the previous problem. I F(a1;a2) = a1a2 − a1 − a2: I Frobenius problem is open for n ≥ 3. ∗ Curtis: it is impossible to find a polynomial formula that computes the Frobenius number if n = 3. ∗ Ram´ırez Alfons´ın: the problem is NP-hard for n variables. A. M. Robles-Perez´ (UGR) Frobenius number and binomial sequences IMNS-2018 2 / 29 Particular cases I Arithmetic and almost arithmetic sequences (A. Brauer; M. Lewin; J. B. Roberts; E. S. Selmer), I Fibonacci sequences (J. M. Mar´ın, J. L. Ram´ırez Alfons´ın and M. P. Revuelta), I geometric sequences (D. C. Ong and V. Ponomarenko), I Mersenne, repunit, and Thabit sequences (J. C. Rosales, M. B. Branco, D. Torrao),˜ I squares and cubes sequences (M. Lepilov, J. O’Rourke and I. Swanson; A. Moscariello), I ... A. M. Robles-Perez´ (UGR) Frobenius number and binomial sequences IMNS-2018 3 / 29 Particular cases: motivations (i) I Brauer: n − 2 F(n;n + 1;:::;n + k − 1) = + 1 n − 1: k − 1 I Baker (conjecture): If Tn is the nth triangular (or triangle) number, then 8 3 2 > 6n +18n +12n−8 if n is even; <> 8 F(Tn;Tn+1;Tn+2) = :> 6n3+12n2−6n−20 8 if n is odd. -
Pascal's Triangle
PASCAL’S TRIANGLE Todd Cochrane 1 / 29 Robert’s Dream In his dream, Robert and the Number Devil build a giant pyramid of blocks. Being two-dimensional they agree to call it a triangle of blocks. Then they start marking the blocks with a felt pen, labeling the top block 1, the next two 1,1, and then each block thereafter the sum of the two values above it. Thus, the third row is 1,2,1, the fourth 1,3,3,1 and so on. 2 / 29 Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ; 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Rule: Each term in Pascal’s triangle is the sum of the two terms above it. Pascal’s triangle is named after Blaise Pascal, who put together many of its properties in 1653. 3 / 29 Triangle Numbers. Diagonal 3: 1; 3; 6; 10; 15; 21;::: Diagonal 4: 1; 4; 10; 20; 35; 56;::: Tetrahedral Numbers. Diagonal’s of Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ; 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Diagonals: Note, the Southwesterly and Southeasterly diagonals are the same. Diagonal 1: 1; 1; 1; 1;::: (yawn) Diagonal 2: 1; 2; 3; 4; 5; : : : ::: Natural Numbers . -
The Personality of the Integers from One to One Gross
ETY CI O O F S The Dozenal Society of America E 0 1 A L X 2 M A 9 3 E The Personality of the Integers N R E 8 4 I Z 7 5 C from One to One Gross A O 6 D Jay Schiffman S Introduction n his fascinating paperback entitled The Lore of integer that lends a special personality to them es- I Large Numbers,1 author Philip J. Davis devotes a pecially when one resorts to duodecimals. Finally, detailed chapter to The Personality of Numbers and my goal is to furnish number theoretic characteris- provides a list entitled Who’s Who Among The Inte- tics related to each of the integers that are perhaps gers from One to One Hundred in the decimal base. omitted from the average list. Our initial task is to The purpose of my paper is to expand his list to the present the reader with a dictionary which is termed first gross and present additional characteristics of A Glossary of Symbols and Terms. These symbols these integers. It is indeed the uniqueness of each and terms will be illustrated in this section. A Glossary of Symbols and Terms 1. n! The factorial of the positive integer n. We de- 6. Ln The nth Lucas Number. n is a Lucas Number fined n!, known as n factorial, recursively as if n satisfied the Lucas Sequence defined recur- follows: 0! = 1 and (n + 1)! = (n + 1)(n!). sively as follows: L1 = 1, L2 = 3, and Ln = Hence 1! = (0 + 1)(0!) = (1)(1) = 1 and Ln−2 + Ln−1 for n ≥ 3. -
Math Ontological Basis of Quas
Math Ontological Basis of Quasi Fine-Tuning in Ghc Cosmologies Mark Thomas To cite this version: Mark Thomas. Math Ontological Basis of Quasi Fine-Tuning in Ghc Cosmologies. 2015. hal- 01232022v5 HAL Id: hal-01232022 https://hal.archives-ouvertes.fr/hal-01232022v5 Preprint submitted on 23 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Math Ontological Basis of Quasi Fine-Tuning in Ghc Cosmologies 1 M. A. Thomas The subject of fine tuning in physics is a long contentious issue especially now as it has hitched a ride on the Multiverse bandwagon. The maths of quadratic forms are predominately featured and relate the physics parameters G h c, which in turn are weighted during the Planck Era(s) determined by relative Planck time clocking. This simplifies the search to these three values as being the important apparent fine-tuned parameters (quasi fine tuning) for determining the gravitational build structures restricted to SM-4D type Universes. Two gravitational coupling constants (dimensionless) are prescribed within the Ghc complex. Both describe the relative rigidity of gravitational physics in the low energy build of our Universe (General Relativity toward endpoint neutron star, black hole formation). -
International Journal of Engineering Research-Online a Peer Reviewed International Journal Vol.1., Issue.3., 2013 Articles Available Online
International journal of Engineering Research-Online A Peer Reviewed International Journal Vol.1., Issue.3., 2013 Articles available online http://www.ijoer.in RESEARCH ARTICLE ISSN: 2321-7758 AN INTERESTING TRANSCENDENTAL EQUATION WITH SIX UNKNOWNS 3 2 x2 y 2 xy X 2 Y 2 z 2 w 2 M.A.GOPALAN, S.VIDHYALAKSHMI, K.LAKSHMI Department of Mathematics, Shrimati Indira Gandhi College,Trichy-620002. Article Received: 11/11/2013 Article Revised on: 21/11/2013 Article Accepted on: 22/11/2013 ABSTRACT The transcendental equation with six unknowns involving surds represented by the 3 equation 2 x2 y 2 xy X 2 Y 2 z 2 w 2 is analyzed for its patterns of non-zero distinct integral solutions. Infinitely many non-zero integer sextuple (,,,,,)x y X Y z w satisfying the above equation are obtained. Three different patterns for finding the solution to the above problem are discussed. The relations between the solutions and the Polygonal numbers, Pyramidal numbers, Pronic number, Jacobsthal number, Jacobsthal-Lucas number, Octahedral number, kynea K.LAKSHMI number, Centered pyramidal numbers and Four Dimensional Figurative numbers are presented. KEYWORDS: Transcendental equation, integral solutions, the Polygonal numbers, Pyramidal numbers, Pronic number, Jacobsthal number, Jacobsthal-Lucas number, Octahedral number, kynea number, Centered pyramidal numbers and Four Dimensional Figurative numbers. M.Sc 2000 mathematics subject classification: 11D99 NOTATIONS: KYn -kynea number of rank Tmn, -Polygonal number of rank n with size m CPn,3 - Centered Triangular pyramidal number of m Pn - Pyramidal number of rank with size rank CP - Centered hexagonal pyramidal number of PRn - Pronic number of rank n n,6 rank OHn - Octahedral number of rank n F4,n ,3 - Four Dimensional Figurative number of SOn -Stella octangular number of rank rank whose generating polygon is a triangle S -Star number of rank n F4,n ,5 - Four Dimensional Figurative number of Jn -Jacobsthal number of rank of rank whose generating polygon is a pentagon.