1,2

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.

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.

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

Cf. A007565, A048289, A137604-A137607.

Sequence in context: A022917 A107762 A027562 this_sequence A087645 A107761 A147943

Adjacent sequences: A096256 A096257 A096258 this_sequence A096260 A096261 A096262

nonn

Yasutoshi Kohmoto (zbi74583(AT)boat.zero.ad.jp), Aug 01 2004; revised Apr 23 2008