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%I A223541 #36 Jun 08 2020 22:40:50
%S A223541 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,
%T A223541 48,64,128,128,128,128,128,128,128,128,256,192,96,192,24,192,96,192,
%U A223541 256,512,512,176,176,44,44,176,176,512,512,1024,768
%N A223541 Array T(m,n) = nim-product(2^m,2^n) (m>=0, n>=0) read by antidiagonals.
%C A223541 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).
%C A223541 The diagonal is A006017 (nim-squares of powers of 2).
%C A223541 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.)
%C A223541 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
%D A223541 J. H. Conway, "Integral lexicographic codes." Discrete Mathematics 83.2-3 (1990): 219-235. See Table 4.
%H A223541 Tilman Piesk, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EyMjM1NDEvYjIyMzU0MS50eHQ">First 128 rows of the matrix, flattened</a>
%H A223541 Tilman Piesk, <a href="/load/view.php?a=aHR0cDovL2NvbW1vbnMud2lraW1lZGlhLm9yZy93aWtpL0NhdGVnb3J5Ok5pbS1wcm9kdWN0c19vZl8yLXBvd2VycztfZHVhbA">Elements of dual matrix</a> (256 SVGs)
%H A223541 Tilman Piesk, <a href="/load/view.php?a=aHR0cDovL2VuLndpa2l2ZXJzaXR5Lm9yZy93aWtpL1dhbHNoX3Blcm11dGF0aW9uO19uaW1iZXJfbXVsdGlwbGljYXRpb24">Walsh permutation; nimber multiplication</a> (Wikiversity)
%H A223541 Tilman Piesk, Class <a href="/load/view.php?a=aHR0cDovL3Bhc3RlYmluLmNvbS9LMDJ3R3NhZQ">bin</a> and function <a href="/load/view.php?a=aHR0cDovL3Bhc3RlYmluLmNvbS9nN0JpMmF5QQ">nimprod</a> (MATLAB code)
%F A223541 T(m,n) = A051775(A000079(m),A000079(n)).
%F A223541 T(m,n) = A223543(A223542(m,n)).
%e A223541 T(1,7) = T(3,5) = 192, the result of the nim-multiplications 2^1*2^7 and 2^3*2^5.
%e A223541 The array begins:
%e A223541 1,2,4,8,16,32,64,128,256,...
%e A223541 2,3,8,12,32,48,128,192,512,...
%e A223541 4,8,6,11,64,128,96,176,1024,...
%e A223541 8,12,11,13,128,192,176,208,2048,...
%e A223541 16,32,64,128,24,44,75,141,4096,...
%e A223541 32,48,128,192,44,52,141,198,8192,...
%e A223541 64,128,96,176,75,141,103,185,16384,...
%e A223541 128,192,176,208,141,198,185,222,32768,...
%e A223541 256,512,1024,2048,4096,8192,16384,32768,384,...
%e A223541 ...
%o A223541 (MATLAB, see code linked above)
%o A223541 A = bin([256 256],'pre') ;
%o A223541 for m=1:256
%o A223541     for n=1:m
%o A223541         a = nimprod( bin(m-1) , bin(n-1) ) ;
%o A223541         A(m,n) = a ;
%o A223541         A(n,m) = a ;
%o A223541     end
%o A223541 end
%Y A223541 Cf. A051775, A223540, A006017 (main diagonal), A223543, A223542, A000079, A002487.
%Y A223541 For rows 1,2,3,4, see A134683, A335159, A335160, A335161.
%K A223541 nonn,tabl
%O A223541 0,2
%A A223541 _Tilman Piesk_, Mar 21 2013
%E A223541 Edited by _N. J. A. Sloane_, Jun 08 2020

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