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

Offset: 0

Author

Tilman Piesk, Mar 21 2013

Keywords

• nonn
• tabl

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)
• M_7(0..7,0..7) <= KM_4(0..7,0..7)
• However, this row does not continue for the matrices of size 4, 2 and 1.

Crossrefs

Cf. A223537, A223539.

Formula

A223537(m,n) = A223539(a(m,n)).