Nim Matrix

The Sprague-Grundy function of graph (X, F) is a function g: X \rightarrow \mathbb{N} such that

g(x) = \min\{ n \geq 0 : n \neq g(y),\, y\, F(x) \}.

This formula can be written compactly using the concept of $\text{mex}$ (minimal excludant). The minimal excludant of a set of non-negative integers is defined as the smallest non-negative number not belonging to this set. Then

g(x) = \textbf{mex}\{ g(y) : y\, F(x) \}.

Let $S$ consist only of non-negative integers. We define $\text{mex}(S)$ as the smallest element not in $S$. Then the infinite matrix $A$ is defined as: A(0, 0) = 0, A(i, j) = \textbf{mex}(S), where $S$ is given by S = S_1 \cap S_2, S_1 = \{ A(k, j) : k < i \}, S_2 = \{ A(i, k) : k < j \}. Below are the first few values ​​of this matrix:

Your task is to compute the specified elements of this matrix.