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A276168 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.

%I A276168 #13 May 04 2025 06:20:57
%S A276168 1,1,0,0,-3,1,-8,0,-15,1,-24,0,-35,1,-48,0
%N A276168 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 A276168 A positive value indicates that all permutations of 1..n result in a game where the first player has a winning strategy.
%C A276168 A negative value indicates that there exists a permutation of 1..n where the second player has a winning strategy.
%C A276168 a(2*k) is strictly nonnegative.
%C A276168 Conjecture: a(2*n - 1) = -A067998(n).
%C A276168 Conjecture: a(n) = n + 1 - A276163(n + 1) for all n >= 1.
%D A276168 Peter Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters/CRC Press, 2003, pages 1-2.
%e A276168 a(1) = 1;   via [1]
%e A276168 a(2) = 1;   via [1,2]
%e A276168 a(3) = 0;   via [1,3,2]
%e A276168 a(4) = 0;   via [1,2,4,3]
%e A276168 a(5) = -3;  via [1,4,2,5,3]
%e A276168 a(6) = 1;   via [1,2,3,4,6,5]
%e A276168 a(7) = -8;  via [1,5,2,6,3,7,4]
%e A276168 a(8) = 0;   via [1,2,3,4,6,5,8,7]
%e A276168 a(9) = -15; via [1,6,2,7,3,8,4,9,5]
%o A276168 (Haskell)
%o A276168 minimax [] = 0
%o A276168 minimax as = max (head as - minimax (tail as)) (last as - minimax (init as))
%o A276168 a276168 n = minimum $ map minimax $ permutations [1..n]
%Y A276168 Cf. A276163.
%K A276168 sign,more
%O A276168 1,5
%A A276168 _Peter Kagey_, Aug 29 2016
%E A276168 a(11)-a(16) from _Bert Dobbelaere_, May 04 2025