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Pascal's Triangle Modulo 3.

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Title: Pascal's Triangle Modulo 3.
Authors: WILSON, AVERY1
Source: Mathematical Spectrum. 2014/2015, Vol. 47 Issue 2, p72-75. 4p.
Subjects: Pascal's triangle, Binomial coefficients, Fractals, Moduli theory, Coefficients (Statistics)
Abstract: If you colour the odd entries of Pascal's triangle red and the even entries blue, a beautiful fractal pattern known as Sierpinski's gasket appears. A well-known problem is to determine how many odd entries appear in any given row of Pascal's triangle. A natural generalization of this problem is to ask, if we look at the nth row of Pascal's triangle modulo any positive integer m, how many occurrences of each residue c lass 0, 1, 2,..., m-1 will we find? Patterns are hard to come by for composite moduli, but nice formulae can be found for prime moduli. In this article, I derive the solution for the modulo 3 case of the problem using Lucas' theorem on binomial coefficients modulo a prime. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Spectrum is the property of Applied Probability Trust School of Mathematics & Statistics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: If you colour the odd entries of Pascal's triangle red and the even entries blue, a beautiful fractal pattern known as Sierpinski's gasket appears. A well-known problem is to determine how many odd entries appear in any given row of Pascal's triangle. A natural generalization of this problem is to ask, if we look at the nth row of Pascal's triangle modulo any positive integer m, how many occurrences of each residue c lass 0, 1, 2,..., m-1 will we find? Patterns are hard to come by for composite moduli, but nice formulae can be found for prime moduli. In this article, I derive the solution for the modulo 3 case of the problem using Lucas' theorem on binomial coefficients modulo a prime. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mathematical Spectrum is the property of Applied Probability Trust School of Mathematics & Statistics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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      – SubjectFull: Binomial coefficients
        Type: general
      – SubjectFull: Fractals
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      – SubjectFull: Moduli theory
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      – SubjectFull: Coefficients (Statistics)
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      – TitleFull: Pascal's Triangle Modulo 3.
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              Text: 2014/2015
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