GEOMETRIC PATTERN OF POWERS OF THREE VIA SIERPINSKI TRIANGULAR NUMBERS

Year 2024, Volume: 25 Issue: 3 , 485 - 489 , 30.09.2024

Temel Ermiş

https://doi.org/10.18038/estubtda.1491452

https://izlik.org/JA32MC93GZ

Abstract

In this study, we describe a new number sequence based on integers arranged in a fractal like structure to visualize powers of three. Moreover, we associate these new numbers with triangular numbers

Keywords

Triangular number , Sierpinski triangular number , Sierpinski triangle , Diophantine equation

References

• [1] Bruckman P, Dence JB, Dence TP, Young J. Series of reciprocal triangular numbers. The College Math J 2013; 44(3): 177-184.
• [2] Caglayan G. Covering a triangular number with pentagonal numbers. Math Intelligencer 2020; 42: 55.
• [3] Cagman A. Explicit solutions of powers of three as sums of three Pell numbers based on Baker’s type inequalities. Turkish Journal of Inequalities 2021; 5(1): 93-103.
• [4] Coons M, Winning HH. Powers of two modulo powers of three. Journal of Integer Sequences 2015; 18: article 15.6.1.
• [5] Charles FM. Proof without words: Square triangular sums. Mathematics Magazine 2019; 92(4): 269.
• [6] Edgar T. Proof without words: A recursion for triangular numbers and more. Mathematics Magazine 2017; 90(2): 124-125.
• [7] Edgar T. Visual triangular number identities from positional number systems. The College Math J 2021; 52(2): 133-136.
• [8] Hopkins B. Proof without words: Products of odd squares and triangular numbers. Mathematics Magazine 2018; 91(1): 42.
• [9] Jones MA. Proof without words: The square of a balancing number is a triangular number. The College Math J 2012; 43(3): 212.
• [10] Leach CD. Proof without words: Powers of three and triangular numbers. The College Math J 2016; 47: 120.
• [11] Matthew JH, Jones MA. Proof without words: Nonnegative integer solutions and triangular numbers. Mathematics Magazine 2002; 75(5): 388.
• [12] Nelsen RB. Proof without words: Squares of triangular numbers. Mathematics Magazine 1990; 63(3): 178.
• [13] Nelsen RB. Proofs without words II. Classroom Resource Materials. Mathematical Association of America, pp 108, 2000.
• [14] Plaza Á. Proof without words: Sum of triangular numbers. Mathematics Magazine 2016; 89(1): 36-37.
• [15] Stephen LS. Proof without words: Alternating sum of squares = triangular number. Mathematics Magazine 1992; 65(2): 90.
• [16] Tiebekabe P, Diouf I. Powers of three as difference of two Fibonacci numbers. JP Journal of Algebra, Number Theory and Applications 2021; 49(2): 185-196.
• [17] Unal H. A visual proof for the sum of the first n triangular numbers. Math Intelligencer 2010; 32: 6-7.
• [18] Zerger MJ. Proof without words: Sums of triangular numbers. Mathematics Magazine 1990; 63(5): 314.