cp's OEIS Frontend

A nim-multiplication table (A051775) of size 2^2^n can be compressed to a matrix of size 2^n, using Walsh permutations. As the nim-multiplication tables are submatrices to the bigger ones, also the compressions are submatrices to the bigger ones, leading to this infinite array.

This array is closely related to the nim-multiplication table powers of 2 (A223541). Both arrays can be seen as different views of the same cubic binary tensor.

The diagonal is A001317 (Sierpinski triangle rows read like binary numbers).

The elements of this array are listed in A223539. In the key-matrix A223538 the entries of this array (which become very large) are replaced by the corresponding index numbers of A223539. (Surprisingly, the key-matrix seems to be interesting on its own.)

Matrix A223537 has very large entries, which are listed in A223539. This matrix has the same pattern as A223537, but the actual entries are replaced by the index numbers of A223539. Surprisingly, although it is just a helper, the key-matrix is mathematically interesting on its own. (See the fractal patterns in the SVG files of the binary dual matrix.) There is even a connection between the binary digits of the actual matrix (A223537) and its key-matrix: It seems that for all matrices of size 8 or bigger the highest binary digits in the actual matrix are less than or equal to the highest binary digits in the key-matrix. (For technical reasons this is shown in the links section.)