A221146 -id:A221146 - cp's OEIS frontend

A003987 Table of n XOR m (or Nim-sum of n and m) read by antidiagonals with m>=0, n>=0.

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, 13, 13, 1, 1, 13, 13, 9, 9, 13, 13

Offset: 0

Marc LeBrun

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.

			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  ...
   ...

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.

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).
N. J. A. Sloane, The OEIS: A Fingerprint File for Mathematics, arXiv:2105.05111 [math.HO], 2021. Mentions this sequence.
Index entries for sequences related to Nim-sums

Initial rows are A001477, A004442, A004443, A004444, etc. Cf. A051775, A051776.
Cf. A003986 (OR), A004198 (AND), A221146 (carries).
Antidiagonal sums are in A006582.

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

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

tabl(nn) = {for(n=0, nn, for(k=0, n, print1(bitxor(k, n - k),", ");); print(););};
tabl(13) \\ Indranil Ghosh, Mar 31 2017

for n in range(14):
    print([k^(n - k) for k in range(n + 1)]) # Indranil Ghosh, Mar 31 2017

T(2i,2j) = 2T(i,j), T(2i+1,2j) = 2T(i,j) + 1.

A004198 Table of x AND y, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 1, 2, 1, 0, 0, 0, 2, 2, 4, 4, 2, 2, 0, 0, 0, 1, 0, 3, 4, 5, 4, 3, 0, 1, 0, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 2, 1, 0, 5, 6, 5, 0, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 6, 6, 0, 0, 2, 2, 0, 0, 0, 1, 0

Offset: 0

David W. Wilson

Or, table of AND(i,j), i >= 0, j >= 0, read by antidiagonals. - N. J. A. Sloane, Feb 08 2016

Or, table of (i+j-Nimsum(i,j))/2 read by antidiagonals [Winning Ways, p. 75]. - N. J. A. Sloane, Feb 22 2019

			The AND(i,j) table (shown without commas or spaces) begins:
0000000000000000000000000...
0101010101010101010101010...
0022002200220022002200220...
0123012301230123012301230...
0000444400004444000044440...
0101454501014545010145450...
0022446600224466002244660...
0123456701234567012345670...
0000000088888888000000008...
0101010189898989010101018...
...
The first few antidiagonals are:
0,
0, 0,
0, 1, 0,
0, 0, 0, 0,
0, 1, 2, 1, 0,
0, 0, 2, 2, 0, 0,
0, 1, 0, 3, 0, 1, 0,
0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 2, 1, 4, 1, 2, 1, 0,
0, 0, 2, 2, 4, 4, 2, 2, 0, 0,
0, 1, 0, 3, 4, 5, 4, 3, 0, 1, 0,
...
- _N. J. A. Sloane_, Feb 08 2016

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 75.

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

Cf. A003986 (OR) and A003987 (XOR). Cf. also A075173, A075175, A221146.

#include 
int main()
{
int n, k;
for (n=0; n<=20; n++){
    for(k=0; k<=n; k++){
        printf("%d, ", (k&(n - k)));
    }
    printf("\n");
}
return 0;
} /* Indranil Ghosh, Apr 01 2017 */

# Maple code for first M rows and columns of AND(i,j) table
M:=24;
f1:=n->[seq(ANDnos(i,n),i=0..M-1)];
for n from 0 to M-1 do lprint(f1(n)); od:
# N. J. A. Sloane, Feb 08 2016

Table[BitAnd[k, n - k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, Apr 01 2017 *)

tabl(nn) = {for(n=0, nn, for(k=0, n, print1(bitand(k, n - k), ", "); ); print(); ); };
tabl(20) \\ Indranil Ghosh, Apr 01 2017

for n in range(21):
    print([k&(n - k) for k in range(n + 1)])
# Indranil Ghosh, Apr 01 2017

cp's OEIS Frontend

A003987 Table of n XOR m (or Nim-sum of n and m) read by antidiagonals with m>=0, n>=0.

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