A276166 a(n) is the first player's score in a "Coins in a Row" game over the n-th row of A066099 using a minimax strategy.

0, 1, 2, 1, 3, 2, 2, 2, 4, 3, 2, 3, 3, 2, 3, 2, 5, 4, 3, 4, 3, 3, 3, 3, 4, 2, 3, 3, 4, 3, 3, 3, 6, 5, 4, 5, 3, 4, 4, 4, 4, 3, 4, 3, 4, 4, 3, 4, 5, 2, 3, 4, 4, 3, 4, 3, 5, 4, 3, 3, 4, 3, 4, 3, 7, 6, 5, 6, 4, 5, 5, 5, 4, 4, 5, 4, 4, 5, 4, 5, 5, 3, 4, 4, 5, 4, 4

Offset

0,3

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.

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

Formula

a(n) = A029837(n + 1) - A276167(n).

a(n) = A276165(n) + A276167(n).

Example

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)    = 1 via [R,L]
A066099_Row(22)   = [2,1,2];     a(22)   = 3 via [L,R,L]
A066099_Row(88)   = [2,1,4];     a(88)   = 5 via [R,L,L]
A066099_Row(1418) = [2,1,4,2,2]; a(1418) = 5 via [L,R,R,R,L]

Prog

(Haskell)
minimaxDifference [] = 0
minimaxDifference as = max (head as - minimaxDifference (tail as)) (last as - minimaxDifference (init as))
minimaxScore as = (sum as + minimaxDifference as) `div` 2
a276166 = minimaxScore . a066099_row

Crossrefs

Cf. A276163, A276164, A276165, A276167.

Sequence in context: A297167 A325224 A303389 * A177062 A133924 A023135

Adjacent sequences: A276163 A276164 A276165 * A276167 A276168 A276169

Keyword

nonn

Author

Peter Kagey, Aug 25 2016

Status

approved