A395248 Sprague-Grundy values for the octal game .007.

0, 0, 0, 1, 1, 1, 2, 2, 0, 3, 3, 1, 1, 1, 0, 4, 3, 3, 3, 2, 2, 2, 4, 4, 0, 5, 5, 2, 2, 2, 3, 3, 0, 5, 0, 1, 1, 1, 3, 3, 3, 5, 6, 4, 4, 1, 0, 5, 5, 6, 6, 2, 7, 7, 7, 8, 0, 1, 9, 2, 7, 2, 3, 3, 3, 9, 0, 5, 4, 4, 8, 6, 6, 2, 7, 1, 1, 1, 0, 5, 5, 9, 3, 1, 8, 2

Offset

0,7

Comments

In the octal game .007, a move removes exactly 3 tokens from a heap and may leave the remaining tokens as zero, one, or two nonempty heaps.

.007 is one of the unsolved octal games: whether the sequence is eventually periodic is unknown (cf. Flammenkamp's tables and R. K. Guy's list of unsolved problems in combinatorial games).

Values computed through n = 2^30 = 1073741824 by the submitter (2026), extending A. Flammenkamp's computation, which reached 2^28. The last "rare" value in Flammenkamp's sparse-space sense occurs at n = 5029983; no further rare values occur through 2^30. The maximum value through 2^30 is 1689, at n = 248902927.

Links

Tyler Satchel Orden, Table of n, a(n) for n = 0..10000

Achim Flammenkamp, Sprague-Grundy values of octal games.

Example

a(3) = 1: from a heap of 3 the only move removes all three tokens, reaching the zero position, so the Grundy value is mex{0} = 1.

Prog

(Python)
def gvals(N):
    G = [0, 0, 0]
    for n in range(3, N+1):
        s = n - 3
        S = {0} if n == 3 else set()
        if s >= 1: S.add(G[s])
        for i in range(1, s//2 + 1): S.add(G[i] ^ G[s-i])
        m = 0
        while m in S: m += 1
        G.append(m)
    return G

Crossrefs

Cf. A002186 (Kayles), A002187 (Dawson's chess), A002188 (Grundy's game).

Sequence in context: A087508 A095731 A048142 * A071426 A288530 A295675

Adjacent sequences: A395245 A395246 A395247 * A395250 A395251 A395252

Keyword

nonn,new

Author

Tyler Satchel Orden, Jul 12 2026

Status

approved