A276166 -id:A276166 - cp's OEIS frontend

A276165 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.

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, 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, 1, 3, 3, 3, 1, 1, 3, 1, 1, 3, 1, 3, 3, -1, 1, 1, 3, 1

Offset: 0

Peter Kagey, Aug 25 2016

sign

"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.

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

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

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

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

Cf. A276163, A276164, A276166, A276167.

minimax [] = 0
minimax as = max (head as - minimax (tail as)) (last as - minimax (init as))
a276165 = minimax . a066099_row

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

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.

Original entry on oeis.org

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

Peter Kagey, Aug 25 2016

nonn

"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.

			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]

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

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

Cf. A276163, A276164, A276165, A276166.

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

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

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

cp's OEIS Frontend

A276165 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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