A295041 The Grundy number of restricted Nim with a pass move.

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, 7, 27, 12, 26, 2, 29, 15, 28, 6, 31, 14, 30, 1, 33, 17, 32, 9, 35, 16, 34, 5, 37, 19, 36, 8

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

Andrew Howroyd, Table of n, a(n) for n = 0..10000

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}]

Prog

(PARI) lista(n) = {my(a=vector(n+1)); for(n=1, n, my(k=n\4); a[1+n]=if(n%2==0, 2*k+(n%4==0), if(n%4==3, a[2*k+2], k + if(n%8==1, 1, -1)))); a} \\ Andrew Howroyd, Oct 30 2025

Crossrefs

Cf. A025480.

Sequence in context: A165066 A034389 A084196 * A359710 A127913 A135991

Adjacent sequences: A295038 A295039 A295040 * A295042 A295043 A295044

Keyword

nonn

Author

Ryouhei Miyadera, Mariko Kashihara and Koh Oomori, Nov 12 2012

Extensions

a(51) onwards from Andrew Howroyd, Oct 30 2025

Status

approved