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 A003987 Table of n XOR m (or Nim-sum of n and m) read by antidiagonals, i.e. with entries in the order (n,m) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),... 48

%I

%S 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,

%T 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,

%U 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

%N Table of n XOR m (or Nim-sum of n and m) read by antidiagonals, i.e. with entries in the order (n,m) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

%C 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.

%C After a few moves the matrix looks like this:

%C 01234...

%C 10325...

%C 2301?

%C The ? is then replaced by 6.

%D J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.

%D J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

%D 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]

%D R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.

%H T. D. Noe, <a href="/A003987/b003987.txt">Rows n=0..100 of triangle, flattened</a>

%H J.-P. Allouche and J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">The Ring of k-regular Sequences, II</a>

%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).

%H <a href="/index/Ni#Nimsums">Index entries for sequences related to Nim-sums</a>

%F T[2i,2j] = 2T[i,j], T[2i+1,2j] = 2T[i,j] + 1.

%e Table begins

%e 0 1 2 3 4 5 6 7 ...

%e 1 0 3 2 5 4 7 6 ...

%e 2 3 0 1 6 7 4 5 ...

%e 3 2 1 0 7 6 5 4 ...

%e 4 5 6 7 0 1 2 3 ...

%e 5 4 7 6 1 0 3 2 ...

%e 6 7 4 5 2 3 0 1 ...

%e 7 6 5 4 3 2 1 0 ...

%e ...................

%p 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

%p 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:

%p # alternative:

%p A003987 := proc(n,m)

%p XORnos(n,m) ;

%p end proc: # _R. J. Mathar_, Apr 17 2013

%t Flatten[Table[BitXor[b, a - b], {a, 0, 10}, {b, 0, a}]] (* BitXor and Nim Sum are equivalent *)

%Y Initial rows are A001477, A004442, A004443, A004444, etc. Cf. A051775, A051776.

%Y Cf. A003986 (OR) and A004198 (AND).

%Y Antidiagonal sums are in A006582.

%K tabl,nonn,nice

%O 0,4

%A _Marc LeBrun_

%E Corrected and formatted example, Tilman Piesk, May 29 2011.

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