A "number wall" is an infinite grid associated to some (bi-directional) sequence. The "numbers" in the grid are usually either integers, or integers modulo some prime. Number walls can be defined in terms of determinants, but can also be calculated using a recursive algorithm.

Formally, let \( R \) be an integral domain. The number wall of a sequence ( S : mathbb{Z} to R ) is an infinite grid of numbers \( W \), defined as follows:

\[ W(x, y) = \begin{cases} 0 & y < -1 \\ 1 & y = -1 \\ S(x) & y = 0 \\ D(x, y) & y > 0 \end{cases} \]

\[ D(x, y) = \begin{vmatrix} S(x) & S(x + 1) & \ldots & S(x + y) \\ S(x - 1) & S(x) & \ldots & S(x + y - 1) \\ \vdots & \vdots & \ddots & \vdots \\ S(x - y) & S(x - y + 1) & \ldots & S(x) \end{vmatrix} \]

The values in any number wall satisfy the relation ( W(x, y - 1) W(x, y + 1) + W(x - 1, y) W(x + 1, y) = W(x, y)^2 ), and some other more complicated relations. You can use these to define a recursive algorithm for generating number walls.

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See this video for more information and fun facts.

• NumberWall