cp's OEIS Frontend

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

%I A096259 #10 Jan 20 2025 22:41:09
%S A096259 1,2,6,24,70,180,294,112,270,900,330,792
%N A096259 Longest period of an abstract version of the game of Go on a 1 X n board.
%C A096259 Rules: 1. If a set of a player's stones has no "open edge" then the other player get the set of stones.
%C A096259 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 A096259 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 A096259 A board is represented as follows.
%C A096259   + + + +
%C A096259   + o x +
%C A096259   + + + +
%C A096259 "o" means a white stone, "x" means a black stone.
%C A096259 "Open edge" : An edge which has one node without a stone. Example:
%C A096259   + x x +
%C A096259   x o o x
%C A096259   + x x +
%C A096259 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 A096259 Note that the rules do not specify when a player wins, so the game never terminates.
%F A096259 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 A096259 The case n=3:
%e A096259   t 1 2 3 3 4 4 5 6 6 7 7
%e A096259   + x x x x x + x x + x x
%e A096259   + + + x x x + + o o o +
%e A096259   + + o o + o o o o o o +
%e A096259 t=1 and t=7 are the same, so the period is 6.
%e A096259 a(12) = 12 * 2^0 * (12 + 6 + 3 + 10 + 5 + 9 + 7 + 8 + 4 + 2 + 1 - 1) = 792.
%Y A096259 Cf. A007565, A048289, A137604, A137605, A137606, A137607.
%K A096259 nonn,more
%O A096259 1,2
%A A096259 _Yasutoshi Kohmoto_, Aug 01 2004; revised Apr 23 2008