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 A257092 Square array read by antidiagonals: Nimsum function for "Take-or-Break" Nim where a legal move is defined as: 1) Remove a nonzero number of counters from any pile up to the size of the selected pile  OR 2) Split any pile of size greater than one into two nonzero piles (removing no counters from the board). 0
 0, 1, 1, 2, 0, 2, 3, 4, 4, 3, 4, 5, 0, 5, 4, 5, 2, 6, 6, 2, 5, 6, 3, 1, 0, 1, 3, 6, 7, 8, 8, 8, 8, 8, 8, 7, 8, 9, 3, 1, 0, 1, 3, 9, 8, 9, 6, 10, 2, 6, 6, 2, 10, 6, 9, 10, 7, 5, 11, 5, 0, 5, 11, 5, 7, 10, 11, 12, 12, 4, 12, 4, 4, 12, 4, 12, 12, 11, 12, 13, 7, 13, 3, 13, 0, 13, 3, 13, 7, 13, 12, 13, 10, 14, 14, 10, 2, 14, 14, 2, 10, 14, 14, 10, 13, 14, 11, 9, 7, 9, 11, 1, 0, 1, 11, 9, 7, 9, 11, 14 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Observe that many "safe" three-pile positions in Nim (1-2-3, 1-4-5, 2-4-6, etc.) consist of one pile whose size is the sum of the sizes of the other two.  The "Break" move is designed to trivially defeat these positions by breaking the large pile into copies of the other two. This leaves a clear winning position where every pile has a "twin". Like the standard Nimsum function defined in A003987, this relation constitutes an Abelian group over nonnegative integers where every element is its own inverse. LINKS FORMULA NSum(x,y) = Flip(Flip(x) XOR Flip(y)) Where XOR is the bitwise exclusive OR characteristic of A003987. Flip(n) = 0    if n == 0.         = n+1  if n is odd.         = n-1  if n is even. EXAMPLE The square table defining the relation begins: 0  1  2  3  4  5  6  7  8  9   ... 1  0  4  5  2  3  8  9  6  7   ... 2  4  0  6  1  8  3 10  5 12   ... 3  5  6  0  8  1  2 11  4 13   ... 4  2  1  8  0  6  5 12  3 10   ... 5  3  8  1  6  0  4 13  2 11   ... 6  8  3  2  5  4  0 14  1 16   ... 7  9 10 11 12 13 14  0 16  1   ... 8  6  5  4  3  2  1 16  0 14   ... 9  7 12 13 10 11 16  1 14  0   ... .  .  .  .  .  .  .  .  .  . Reading from the table, 1-2-4, 1-3-5 and 2-3-6 are safe positions in Take-or-Break Nim. PROG (PARI) flip(x) = if (x==0, 0, if (x % 2, x+1, x-1)); tabl(nn) = {for (n=0, nn, for (k=0, nn, print1(flip(bitxor(flip(n), flip(k))), ", "); ); print(); ); } \\ Michel Marcus, Apr 23 2015 CROSSREFS Cf. A003987. Sequence in context: A141693 A279679 A261096 * A028376 A059235 A261216 Adjacent sequences:  A257089 A257090 A257091 * A257093 A257094 A257095 KEYWORD nonn,tabl AUTHOR Patrick McKinley, Apr 19 2015 STATUS approved

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Last modified January 20 21:36 EST 2019. Contains 319336 sequences. (Running on oeis4.)