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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), ... 156
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 [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.

REFERENCES

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.

LINKS

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

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

Rémy Sigrist, Colored representation of T(x,y) for x = 0..1023 and y = 0..1023 (where the hue is function of T(x,y) and black pixels correspond to zeros)

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,  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  ...

   ...

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 *)

PROG

(PARI) tabl(nn) = {for(n=0, nn, for(k=0, n, print1(bitxor(k, n - k), ", "); ); print(); ); };

tabl(13) \\ Indranil Ghosh, Mar 31 2017

(Python)

for n in xrange(0, 14):

....print [k^(n - k) for k in xrange(0, n + 1)] # Indranil Ghosh, Mar 31 2017

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: A323212 A185815 A321132 * A307302 A307297 A307301

Adjacent sequences:  A003984 A003985 A003986 * A003988 A003989 A003990

KEYWORD

tabl,nonn,nice,look

AUTHOR

Marc LeBrun

STATUS

approved

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Last modified October 19 11:09 EDT 2019. Contains 328216 sequences. (Running on oeis4.)