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Longest period of an abstract version of the game of Go on a 1 X n board.
3

%I #5 Apr 19 2016 01:07:34

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

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

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

%C 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.

%C 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.

%C A board is represented as follows.

%C + + + +

%C + o x +

%C + + + +

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

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

%C + x x +

%C x o o x

%C + x x +

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

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

%F 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.

%e The case n=3:

%e t 1 2 3 3 4 4 5 6 6 7 7

%e + x x x x x + x x + x x

%e + + + x x x + + o o o +

%e + + o o + o o o o o o +

%e t=1 and t=7 are the same, so the period is 6.

%e a(12) = 12*2^0*(12 +6 +3 +10 +5 +9 +7 +8 +4 +2 +1 -1) =792

%Y Cf. A007565, A048289, A137604-A137607.

%K nonn

%O 1,2

%A _Yasutoshi Kohmoto_, Aug 01 2004; revised Apr 23 2008