page_title: (2,1)-Pascal triangle
section_title: Overall patterns and properties
hierarchical_section_title: (2,1)-Pascal triangle / Patterns and properties / Overall patterns and properties
caption_attribution_description: English: Sister Pascal's triangle overlaid on a grid gives the number of distinct paths to each square, assuming only rightward and downward movements are considered.
context_page_description: In mathematics, the (2,1)-Pascal triangle is a triangular array.
  The rows of the (2,1)-Pascal triangle are conventionally enumerated starting with row n = 0 at the top. The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows.
  The triangle is based on the Pascal's Triangle with the second line being (2,1) and the first cell of each row set to 2.
  This construction is related to the binomial coefficients by Pascal's rule, with one of the terms being.
context_section_description: The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the fractal called the Sierpinski triangle. This resemblance becomes more and more accurate as more rows are considered; in the limit, as the number of rows approaches infinity, the resulting pattern is the Sierpinski 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.
  Imagine each number in the triangle is a node in a grid which is connected to the adjacent numbers above and below it. Now for any node in the grid, count the number of paths there are in the grid (without backtracking) which connect this node to the top node (1) of the triangle. The answer is the Pascal number associated to that node.
  One property of the triangle is revealed if the rows are left-justified. In the triangle below, the diagonal coloured bands sum to successive Fibonacci numbers and Lucas numbers.
  This construction is also related to the expansion of  , using .
  then
caption: (2,1)-Pascal triangle overlaid on a grid gives the number of distinct paths to each square, assuming only rightward and downward movements are considered.
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