A276167 a(n) is the second player's score in a "Coins in a Row" game over the n-th row of A066099 using a minimax strategy.

0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 1, 2, 0, 1, 2, 1, 2, 2, 2, 2, 1, 3, 2, 2, 1, 2, 2, 2, 0, 1, 2, 1, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 1, 4, 3, 2, 2, 3, 2, 3, 1, 2, 3, 3, 2, 3, 2, 3, 0, 1, 2, 1, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 2, 4, 3, 3, 2, 3, 3

Offset: 0

Author

Peter Kagey, Aug 25 2016

Keywords

• nonn

Comments

"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.

Examples

			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.
A066099_Row(0)    = [0];         a(0)    = 0 via [L]
A066099_Row(1)    = [1];         a(1)    = 0 via [L]
A066099_Row(3)    = [1,1];       a(3)    = 1 via [R,L]
A066099_Row(22)   = [2,1,2];     a(22)   = 2 via [L,R,L]
A066099_Row(88)   = [2,1,4];     a(88)   = 2 via [R,L,L]
A066099_Row(1418) = [2,1,4,2,2]; a(1418) = 6 via [L,R,R,R,L]
		

References

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

Links

• Peter Kagey, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A276163, A276164, A276165, A276166.

Programs

Haskell

minimaxDifference [] = 0
minimaxDifference as = max (head as - minimaxDifference (tail as)) (last as - minimaxDifference (init as))
minimaxScore2 as = (sum as - minimaxDifference as) `div` 2
a276167 = minimaxScore2 . a066099_row

Formula

a(n) = A029837(n + 1) - A276166(n).

a(n) = A276166(n) - A276165(n).