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a(n) is the minimum first-player score difference of a "Coins in a Row" game over all permutations of coins 1..n with both players using a minimax strategy.
1

%I #9 Sep 03 2016 17:10:12

%S 1,1,0,0,-3,1,-8,0,-15,1

%N a(n) is the minimum first-player score difference of a "Coins in a Row" game over all permutations of coins 1..n with both players using a minimax strategy.

%C A positive value indicates that all permutations of 1..n result in a game where the first player has a winning strategy.

%C A negative value indicates that there exists a permutation of 1..n where the second player has a winning strategy.

%C a(2*k) is strictly nonnegative.

%C Conjecture: a(2*n - 1) = -A067998(n).

%C Conjecture: a(n) = n + 1 - A276163(n + 1) for all n >= 1.

%D Peter Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters/CRC Press, 2003, pages 1-2.

%e a(1) = 1; via [1]

%e a(2) = 1; via [1,2]

%e a(3) = 0; via [1,3,2]

%e a(4) = 0; via [1,2,4,3]

%e a(5) = -3; via [1,4,2,5,3]

%e a(6) = 1; via [1,2,3,4,6,5]

%e a(7) = -8; via [1,5,2,6,3,7,4]

%e a(8) = 0; via [1,2,3,4,6,5,8,7]

%e a(9) = -15; via [1,6,2,7,3,8,4,9,5]

%o (Haskell)

%o minimax [] = 0

%o minimax as = max (head as - minimax (tail as)) (last as - minimax (init as))

%o a276168 n = minimum $ map minimax $ permutations [1..n]

%Y Cf. A276163.

%K sign,more

%O 1,5

%A _Peter Kagey_, Aug 29 2016