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A295041 The Grundy number of restricted Nim with a pass move. 0
0, 1, 0, 1, 3, 0, 2, 1, 5, 3, 4, 0, 7, 2, 6, 1, 9, 5, 8, 3, 11, 4, 10, 0, 13, 7, 12, 2, 15, 6, 14, 1, 17, 9, 16, 5, 19, 8, 18, 3, 21, 11, 20, 4, 23, 10, 22, 0, 25, 13, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
These are the Grundy values or nim-values for heaps of n beans in the game where you're allowed to take up to half of the beans in a heap and you can use a one-time pass, i.e., a pass move which may be used at most once in a game, and not from a terminal position. Once the pass has been used by either player, it is no longer available. If the pass move were not allowed, then this game would be the same as the one in A025480.
LINKS
FORMULA
a(4k) = 2k+1; a(4k+2) = 2k; a(4k+3) = a(2k+1); a(8k+1) = 2k+1; a(8k+5) = 2k.
MATHEMATICA
f[n_] := Which[IntegerQ[n/4], (n + 2)/2, IntegerQ[(n - 2)/4], (n - 2)/2,
IntegerQ[(n - 3)/4], f[(n - 1)/2], IntegerQ[(n - 1)/8], (n + 3)/4,
IntegerQ[(n - 5)/8], (n - 5)/4];
(* the following is Mathematica program to generate the same sequence as Grundy numbers *)
ss = 50; allcases = Flatten[Table[Table[{a, pass}, {a, 0, ss}], {pass, 0, 1}], 1];
move[z_] := Block[{p}, p = z;
a = p[[1]]; pass = p[[2]]; c0 = Floor[a/2];
Which[a > 0 && pass == 1,
Union[Table[{a - x, pass}, {x, 1, c0}], {{a, 0}}], a > 0,
Table[{a - x, pass}, {x, 1, c0}], a == 0, {}]];
Mex[L_] := Min[Complement[Range[0, Length[L]], L]];
Gr2[pos_] := Gr2[pos] = Mex[Map[Gr2, move[pos]]];
pposition = Select[allcases, Gr2[#] == 0 &];
Table[Gr2[{n, 1}], {n, 0, 50}]
CROSSREFS
Cf. A025480.
Sequence in context: A165066 A034389 A084196 * A359710 A127913 A135991
KEYWORD
nonn
AUTHOR
Ryouhei Miyadera, Mariko Kashihara and Koh Oomori, Nov 12 2012
STATUS
approved

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Last modified April 23 07:16 EDT 2024. Contains 371905 sequences. (Running on oeis4.)