A096259 Longest period of an abstract version of the game of Go on a 1 X n board.

1, 2, 6, 24, 70, 180, 294, 112, 270, 900, 330, 792

Offset

1,2

Comments

Rules: 1. If a set of a player's stones has no "open edge" then the other player get the set of stones.

2. If the sets of both player's stones has no "open edge" in a configuration, then a player who made this configuration get the set of the other player's stone.

3. A player never make a configuration in which his stones have no open edge and the other player's stones have an open edge.

A board is represented as follows.

+ + + +

+ o x +

+ + + +

"o" means a white stone, "x" means a black stone.

"Open edge" : An edge which has one node without a stone. Example:

+ x x +

x o o x

+ x x +

The center set of white stones has no "open edge", so black player gets them. Six black stones have "open edges" like this : "x +".

Note that the rules do not specify when a player wins, so the game never terminates.

Links

Table of n, a(n) for n=1..12.

Formula

For 4<=n, a(n) = n * 2^p * ( Sum_{0<=k<=m} ( Sum_{0<=i<=h_k} n_k/2^i ) - 1 ) where p = m Mod 2, n_0 = n, n_k = n - [n_{k-1}/2^(h_{k-1}+1)] - 1, 2^h_k is the highest power of two dividing n_k: n_m/2^h_m = 1.

Example

The case n=3:
  t 1 2 3 3 4 4 5 6 6 7 7
  + x x x x x + x x + x x
  + + + x x x + + o o o +
  + + o o + o o o o o o +
t=1 and t=7 are the same, so the period is 6.
a(12) = 12 * 2^0 * (12 + 6 + 3 + 10 + 5 + 9 + 7 + 8 + 4 + 2 + 1 - 1) = 792.

Crossrefs

Cf. A007565, A048289, A137604, A137605, A137606, A137607.

Sequence in context: A253901 A027562 A236625 * A338614 A087645 A174700

Adjacent sequences: A096256 A096257 A096258 * A096260 A096261 A096262

Keyword

nonn,more

Author

Yasutoshi Kohmoto, Aug 01 2004; revised Apr 23 2008

Status

approved