A223538 - OEIS

A223538

Key-matrix of compressed nim-multiplication table (A223537) read by antidiagonals.

0, 1, 1, 3, 2, 5, 5, 5, 7, 7, 9, 7, 4, 3, 25, 11, 11, 6, 6, 15, 15, 15, 13, 20, 8, 22, 20, 28, 20, 20, 25, 25, 28, 28, 17, 17, 30, 25, 17, 15, 10, 17, 19, 22, 68, 32, 32, 22, 22, 12, 12, 24, 24, 86, 86, 36, 34, 40, 28, 16, 14, 21, 27, 90, 104

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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.)

LINKS

Tilman Piesk, First 128 rows of the matrix, flattened

Tilman Piesk, 256x256 key-matrix

Tilman Piesk, Elements of dual matrix (15 SVGs)

Tilman Piesk, Walsh permutation; nimber multiplication (Wikiversity)

Connection between binary digits of A223537 (M) and the key matrix (KM):

Let M_n (KM_n) denote the matrix of binary digits with exponent n in matrix M (KM).

M_255(0..255,0..255) <= KM_14(0..255,0..255)

M_127(0..127,0..127) <= KM_12(0..127,0..127)

M_63(0..63,0..63) <= KM_10(0..63,0..63)

M_31(0..31,0..31) <= KM_8(0..31,0..31)

M_15(0..15,0..15) <= KM_6(0..15,0..15)