Pascal's triangle: Difference between revisions

Revision as of 09:15, 10 February 2025

Revision as of 10:42, 10 February 2025

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=== Overall patterns and properties ===

[[File:Sierpinski Pascal triangle.svg|thumb|A level-4 approximation to a Sierpinski[[Sierpiński triangle]] obtained by shading the first 32 rows of a Pascal triangle white if the binomial coefficient is even and black if it is odd.]]

* The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the [[fractal]] known as the [[SierpinskiSierpiński triangle]]. This resemblance becomes increasingly accurate as more rows are considered; in the limit, as the number of rows approaches infinity, the resulting pattern ''is'' the SierpinskiSierpiński triangle, assuming a fixed perimeter. More generally, numbers could be colored differently according to whether or not they are multiples of 3, 4, etc.; this results in other similar patterns.

:As the proportion of black numbers tends to zero with increasing ''n'', a corollary is that the proportion of odd binomial coefficients tends to zero as ''n'' tends to infinity.<ref>Ian Stewart, "How to Cut a Cake", Oxford University Press, page 180</ref>

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