OFFSET
0,5
COMMENTS
Consider a two-player stone-throwing game with a single shared pile of stones. The players alternately remove one or more stones from the pile until it is empty. In addition, each player seeks to communicate a message through their sequence of moves. If there are initially n stones then a(n) is the largest number m such that both players can communicate at least m distinct messages.
For n > 0, a(n) is also the size of the Durfee square of the partition defined in A064660.
LINKS
Peter J. Taylor, Table of n, a(n) for n = 0..60
Peter J. Taylor, The lapidary numbers, or the combinatorics of communication by throwing stones (preprint).
Peter J. Taylor, The lapidary numbers, or the combinatorics of communication by throwing stones, Eureka, 65 (2018), pp. 89-90.
Peter J. Taylor, Python program
FORMULA
Asymptotically, a(n) is within a subexponential factor of 2^(n/2).
EXAMPLE
For n=4, one strategy which allows both players to communicate one of two messages is each remove one or two stones on their first turn.
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter J. Taylor, Feb 22 2020
STATUS
approved