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a(n) is the first-player score difference of a "Coins in a Row" game over the n-th row of A066099 using a minimax strategy.
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%I #23 Aug 27 2016 02:11:23

%S 0,1,2,0,3,1,1,1,4,2,0,2,2,0,2,0,5,3,1,3,1,1,1,1,3,-1,1,1,3,1,1,1,6,4,

%T 2,4,0,2,2,2,2,0,2,0,2,2,0,2,4,-2,0,2,2,0,2,0,4,2,0,0,2,0,2,0,7,5,3,5,

%U 1,3,3,3,1,1,3,1,1,3,1,3,3,-1,1,1,3,1

%N a(n) is the first-player score difference of a "Coins in a Row" game over the n-th row of A066099 using a minimax strategy.

%C "Coins in a Row" is a game in which players alternate picking up coins of varying denominations from the end of the row in an attempt to collect as many points as possible.

%C When a(n) is negative, the second player has a strategy that is guaranteed to collect more points.

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

%H Peter Kagey, <a href="/A276165/b276165.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = A276166(n) - A276167(n).

%e Let [R,L,L,L] represent a game in which the first player takes the right coin, the second player takes the left coin, the first player takes the left coin, and the second player takes the left (only remaining) coin.

%e A066099_Row(0) = [0]; a(0) = 0 via [L]

%e A066099_Row(1) = [1]; a(1) = 1 via [L]

%e A066099_Row(3) = [1,1]; a(3) = 0 via [R,L]

%e A066099_Row(22) = [2,1,2]; a(22) = 1 via [L,R,L]

%e A066099_Row(88) = [2,1,4]; a(88) = 3 via [R,L,L]

%e A066099_Row(1418) = [2,1,4,2,2]; a(1418) = -1 via [L,R,R,R,L]

%o (Haskell)

%o minimax [] = 0

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

%o a276165 = minimax . a066099_row

%Y Cf. A276163, A276164, A276166, A276167.

%K sign

%O 0,3

%A _Peter Kagey_, Aug 25 2016