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A003987
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Table of n XOR m (or Nim-sum of n and m) read by antidiagonals with m>=0, n>=0.
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199
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0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 0, 2, 4, 5, 5, 1, 1, 5, 5, 6, 4, 6, 0, 6, 4, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 6, 4, 6, 0, 6, 4, 6, 8, 9, 9, 5, 5, 1, 1, 5, 5, 9, 9, 10, 8, 10, 4, 2, 0, 2, 4, 10, 8, 10, 11, 11, 11, 11, 3, 3, 3, 3, 11, 11, 11, 11, 12, 10, 8, 10, 12, 2, 0, 2, 12, 10, 8, 10, 12, 13, 13, 9, 9
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OFFSET
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0,4
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COMMENTS
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Another way to construct the array: construct an infinite square matrix starting in the top left corner using the rule that each entry is the smallest nonnegative number that is not in the row to your left or in the column above you.
After a few moves the [symmetric] matrix looks like this:
0 1 2 3 4 5 ...
1 0 3 2 5 ...
2 3 0 1 ?
3 2 1
4 5 ?
5
The ? is then replaced with a 6.
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REFERENCES
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E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, date?
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see p. 190. [From N. J. A. Sloane, Jul 14 2009]
R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.
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LINKS
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FORMULA
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T(2i,2j) = 2T(i,j), T(2i+1,2j) = 2T(i,j) + 1.
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EXAMPLE
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Table begins
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, ...
2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, ...
3, 2, 1, 0, 7, 6, 5, 4, 11, 10, ...
4, 5, 6, 7, 0, 1, 2, 3, 12, ...
5, 4, 7, 6, 1, 0, 3, 2, ...
6, 7, 4, 5, 2, 3, 0, ...
7, 6, 5, 4, 3, 2, ...
8, 9, 10, 11, 12, ...
9, 8, 11, 10, ...
10, 11, 8, ...
11, 10, ...
12, ...
...
The first few antidiagonals are
0;
1, 1;
2, 0, 2;
3, 3, 3, 3;
4, 2, 0, 2, 4;
5, 5, 1, 1, 5, 5;
6, 4, 6, 0, 6, 4, 6;
7, 7, 7, 7, 7, 7, 7, 7;
8, 6, 4, 6, 0, 6, 4, 6, 8;
9, 9, 5, 5, 1, 1, 5, 5, 9, 9;
10, 8, 10, 4, 2, 0, 2, 4, 10, 8, 10;
11, 11, 11, 11, 3, 3, 3, 3, 11, 11, 11, 11;
12, 10, 8, 10, 12, 2, 0, 2, 12, 10, 8, 10, 12;
...
[Symmetric] matrix in base 2:
0 1 10 11 100 101, 110 111 1000 1001 1010 1011 ...
1 0 11 10 101 100, 111 110 1001 1000 1011 ...
10 11 0 1 110 111, 100 101 1010 1011 ...
11 10 1 0 111 110, 101 100 1011 ...
100 101 110 111 0 1 10 11 ...
101 100 111 110 1 0 11 ...
110 111 100 101 10 11 ...
111 110 101 100 11 ...
1000 1001 1010 1011 ...
1001 1000 1011 ...
1010 1011 ...
1011 ...
...
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MAPLE
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nimsum := proc(a, b) local t1, t2, t3, t4, l; t1 := convert(a+2^20, base, 2); t2 := convert(b+2^20, base, 2); t3 := evalm(t1+t2); map(x->x mod 2, t3); t4 := convert(evalm(%), list); l := convert(t4, base, 2, 10); sum(l[k]*10^(k-1), k=1..nops(l)); end; # memo: adjust 2^20 to be much bigger than a and b
AT := array(0..N, 0..N); for a from 0 to N do for b from a to N do AT[a, b] := nimsum(a, b); AT[b, a] := AT[a, b]; od: od:
# alternative:
read("transforms") :
XORnos(n, m) ;
seq(seq(Bits:-Xor(k, m-k), k=0..m), m=0..20); # Robert Israel, Dec 31 2015
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MATHEMATICA
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Flatten[Table[BitXor[b, a - b], {a, 0, 10}, {b, 0, a}]] (* BitXor and Nim Sum are equivalent *)
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PROG
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(PARI) tabl(nn) = {for(n=0, nn, for(k=0, n, print1(bitxor(k, n - k), ", "); ); print(); ); };
(Python)
for n in range(14):
print([k^(n - k) for k in range(n + 1)]) # Indranil Ghosh, Mar 31 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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