

A223541


Array T(m,n) = nimproduct(2^m,2^n) (m>=0, n>=0) read by antidiagonals.


11



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
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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 (nimsquares of powers of 2).
The elements of this array are listed in A223543. In the keymatrix A223542 the entries of this array (which become very large) are replaced by the corresponding index numbers of A223543. (Surprisingly, the keymatrix 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 nimproduct(2^a,2^b) with a+b=n.  Tilman Piesk, Mar 27 2013


REFERENCES

J. H. Conway, "Integral lexicographic codes." Discrete Mathematics 83.23 (1990): 219235. 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 nimmultiplications 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(m1) , bin(n1) ) ;
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



