# The remarkable properties of Pascal’s triangle

However, the entries in Pascal’s triangle also follow some more obvious patterns: the diagonals of the triangle, which run parallel to the outer rows of the one, list the natural numbers, for example. The reason for this is that the numbers in the next row are always made up of the sum of the two values ​​above: because there is always one number on the outer edge, the sequence increases by one each step.

The diagonals immediately adjacent to the values ​​1, 3, 6, 10, 15, etc. are also interesting. These are triangular numbers: the number of stones you need to be able to put an equilateral triangle out of. This episode appears because nThe third triangle number for the sum of 1 + 2 + … + n equivalent. This can be done according to Gaussian sum formula Through n(nCalculate +1) / 2 – and that’s just about the value B(n+1, 2), i.e. getting into n+ first row and second column of Pascal’s triangle. triangular numbers | This is the number of points needed to build an equilateral triangle.

Another property related to triangles can be seen by coloring the odd and even entries of Pascal’s triangle differently. As François-Edouard Anatole Lucas proved in 1890, this creates a pattern that may sound familiar to some: it’s a fractal called the Sierpinski Triangle. This can be constructed by dividing an equilateral triangle into four smaller congruent triangles and removing the middle triangle. Repeating this process over and over with the three remaining smaller triangles creates a fractal pattern that can also be observed in Pascal’s triangle.