A223541 Array T(m,n) = nim-product(2^m,2^n) (m>=0, n>=0) read by antidiagonals.

1, 2, 2, 4, 3, 4, 8, 8, 8, 8, 16, 12, 6, 12, 16, 32, 32, 11, 11, 32, 32, 64, 48, 64, 13, 64, 48, 64, 128, 128, 128, 128, 128, 128, 128, 128, 256, 192, 96, 192, 24, 192, 96, 192, 256, 512, 512, 176, 176, 44, 44, 176, 176, 512, 512, 1024, 768

Offset: 0

Author

Tilman Piesk, Mar 21 2013

Keywords

• nonn
• tabl

Comments

Nimber multiplication is commutative, so this array is symmetric, and can be represented in a more compact way by the rows of the lower triangle (A223540).

The diagonal is A006017 (nim-squares of powers of 2).

The elements of this array are listed in A223543. In the key-matrix A223542 the entries of this array (which become very large) are replaced by the corresponding index numbers of A223543. (Surprisingly, the key-matrix seems to be interesting on its own.)

The number of different entries per antidiagonal is probably A002487. That would mean that there are exactly A002487(n+1) numbers that can be expressed as a nim-product(2^a,2^b) with a+b=n. - Tilman Piesk, Mar 27 2013

Examples

			T(1,7) = T(3,5) = 192, the result of the nim-multiplications 2^1*2^7 and 2^3*2^5.
The array begins:
1,2,4,8,16,32,64,128,256,...
2,3,8,12,32,48,128,192,512,...
4,8,6,11,64,128,96,176,1024,...
8,12,11,13,128,192,176,208,2048,...
16,32,64,128,24,44,75,141,4096,...
32,48,128,192,44,52,141,198,8192,...
64,128,96,176,75,141,103,185,16384,...
128,192,176,208,141,198,185,222,32768,...
256,512,1024,2048,4096,8192,16384,32768,384,...
...
		

References

• J. H. Conway, "Integral lexicographic codes." Discrete Mathematics 83.2-3 (1990): 219-235. See Table 4.

Links

• Tilman Piesk, First 128 rows of the matrix, flattened
• Tilman Piesk, Elements of dual matrix (256 SVGs)
• Tilman Piesk, Walsh permutation; nimber multiplication (Wikiversity)
• Tilman Piesk, Class bin and function nimprod (MATLAB code)

Crossrefs

Cf. A051775, A223540, A006017 (main diagonal), A223543, A223542, A000079, A002487.

For rows 1,2,3,4, see A134683, A335159, A335160, A335161.

Formula

T(m,n) = A051775(A000079(m),A000079(n)).

T(m,n) = A223543(A223542(m,n)).

Extensions

Edited by N. J. A. Sloane, Jun 08 2020