|
%I #16 Oct 30 2022 15:09:09
%S 1,1,1,1,2,2,3,4,6,8,12,16,23,31,45,61,87,119,171,233,334,459,655,904,
%T 1288,1782,2535,3517,4995,6935,9848,13703,19437,27070,38376,53528,
%U 75842,105878,149966,209555,296707,414922,587304,821853,1163052,1628574,2304082
%N Lapidary numbers.
%C 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.
%C For n > 0, a(n) is also the size of the Durfee square of the partition defined in A064660.
%H Peter J. Taylor, <a href="/A332755/b332755.txt">Table of n, a(n) for n = 0..60</a>
%H Peter J. Taylor, <a href="http://www.cheddarmonk.org/papers/lapidary.pdf">The lapidary numbers, or the combinatorics of communication by throwing stones</a> (preprint).
%H Peter J. Taylor, <a href="https://archim.org.uk/eureka/archive/Eureka-65.pdf">The lapidary numbers, or the combinatorics of communication by throwing stones</a>, Eureka, 65 (2018), pp. 89-90.
%H Peter J. Taylor, <a href="/A332755/a332755.py.txt">Python program</a>
%F Asymptotically, a(n) is within a subexponential factor of 2^(n/2).
%e 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.
%Y Cf. A064660.
%K nonn
%O 0,5
%A _Peter J. Taylor_, Feb 22 2020
|
