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GEOMETRIC PATTERN OF POWERS OF THREE VIA SIERPINSKI TRIANGULAR NUMBERS

Year 2024, , 485 - 489, 30.09.2024
https://doi.org/10.18038/estubtda.1491452

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

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.
Year 2024, , 485 - 489, 30.09.2024
https://doi.org/10.18038/estubtda.1491452

Abstract

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.
There are 18 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Articles
Authors

Temel Ermiş 0000-0003-4430-5271

Publication Date September 30, 2024
Submission Date May 28, 2024
Acceptance Date July 30, 2024
Published in Issue Year 2024

Cite

AMA Ermiş T. GEOMETRIC PATTERN OF POWERS OF THREE VIA SIERPINSKI TRIANGULAR NUMBERS. Estuscience - Se. September 2024;25(3):485-489. doi:10.18038/estubtda.1491452