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
Examples
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 ...
...
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).
- 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
Crossrefs
Programs
-
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 *) -
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 range(14): print([k^(n - k) for k in range(n + 1)]) # Indranil Ghosh, Mar 31 2017
Formula
T(2i,2j) = 2T(i,j), T(2i+1,2j) = 2T(i,j) + 1.
Comments