Search: id:A003987
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%I A003987
%S A003987 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,
%T A003987 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,
%U A003987 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 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),...
%C A003987 T[2i,2j] = 2T[i,j], T[2i+1,2j] = 2T[i,j] + 1.
%D A003987 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. 
               Computer Sci., 307 (2003), 3-29.
%D A003987 J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
%D A003987 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 A003987 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 A003987 T. D. Noe, <a href="b003987.txt">Rows n=0..100 of triangle, flattened</
               a>
%H A003987 J.-P. Allouche and J. Shallit, <a href="http://www.lri.fr/~allouche/kreg2.ps">
               The Ring of k-regular Sequences, II</a>
%H A003987 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A003987 <a href="Sindx_Ni.html#Nimsums">Index entries for sequences related to 
               Nim-sums</a>
%e A003987 Table begins
%e A003987 0 1 2 3 4 5 6 7 ...
%e A003987 1 0 3 2 5 4 7 6 ...
%e A003987 2 3 0 1 6 7 4 5 ...
%e A003987 3 2 1 0 7 6 5 3 ...
%e A003987 4 5 6 7 0 1 2 3 ...
%e A003987 ...................
%p A003987 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 A003987 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:
%t A003987 Flatten[Table[BitXor[b, a - b], {a, 0, 10}, {b, 0, a}]] (BitXor and Nim 
               Sum are equivalent)
%Y A003987 Initial rows are A001477, A004442, A004443, A004444, etc. Cf. A051775, 
               A051776.
%Y A003987 Cf. A003986 (OR) and A004198 (AND).
%Y A003987 Antidiagonal sums are in A006582.
%Y A003987 Sequence in context: A074660 A002125 A135356 this_sequence A141692 A063180 
               A141693
%Y A003987 Adjacent sequences: A003984 A003985 A003986 this_sequence A003988 A003989 
               A003990
%K A003987 tabl,nonn,nice
%O A003987 0,4
%A A003987 Marc LeBrun (mlb(AT)well.com)

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