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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), ... 76
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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 matrix looks like this:

0 1 2 3 4 ...

1 0 3 2 5 ...

2 3 0 1 ?

The ? is then replaced with a 6.

REFERENCES

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

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

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.

LINKS

T. D. Noe, Rows n=0..100 of triangle, flattened

J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Index entries for sequences related to Nim-sums

FORMULA

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

EXAMPLE

Table begins

  0 1 2 3 4 5 6 7 ...

  1 0 3 2 5 4 7 6 ...

  2 3 0 1 6 7 4 5 ...

  3 2 1 0 7 6 5 4 ...

  4 5 6 7 0 1 2 3 ...

  5 4 7 6 1 0 3 2 ...

  6 7 4 5 2 3 0 1 ...

  7 6 5 4 3 2 1 0 ...

  ...................

MAPLE

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") :

A003987 := proc(n, m)

    XORnos(n, m) ;

end proc: # R. J. Mathar, Apr 17 2013

seq(seq(Bits:-Xor(k, m-k), k=0..m), m=0..20); # Robert Israel, Dec 31 2015

MATHEMATICA

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

CROSSREFS

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

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

Antidiagonal sums are in A006582.

Sequence in context: A002125 A171731 A185815 * A141692 A261097 A261217

Adjacent sequences:  A003984 A003985 A003986 * A003988 A003989 A003990

KEYWORD

tabl,nonn,nice,look

AUTHOR

Marc LeBrun

EXTENSIONS

Example corrected and formatted by Tilman Piesk, May 29 2011

STATUS

approved

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Last modified August 30 04:31 EDT 2016. Contains 275965 sequences.