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

0,2

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

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)

Formula

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

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

Example

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,...
...

Prog

(MATLAB, see code linked above)
A = bin([256 256], 'pre') ;
for m=1:256
    for n=1:m
        a = nimprod( bin(m-1) , bin(n-1) ) ;
        A(m, n) = a ;
        A(n, m) = a ;
    end
end

Crossrefs

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

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

Sequence in context: A227256 A328774 A131816 * A128181 A125185 A274895

Adjacent sequences: A223538 A223539 A223540 * A223542 A223543 A223544

Keyword

nonn,tabl

Author

Tilman Piesk, Mar 21 2013

Extensions

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

Status

approved