%I #14 Jul 23 2016 02:07:18
%S 1,1,1,2,4,252,2304,343712160,17361257184,817232021393069622912,
%T 337615438409845853800240,1002314950534089781700930298791626826040504,
%U 4687493998578314511363173974007271386258869456
%N Number of ways to relax the chip-configuration of Anderson et al. one chip at a time.
%C Terms grow cubic-exponentially but have unexpectedly many small prime factors.
%H R. Anderson, L. Lovász, P. Shor, J. Spencer, E. Tardos, S. Winograd, <a href="http://www.jstor.org/stable/2323970">Disks, balls and walls: analysis of a combinatorial game</a>, Amer. Math. Monthly, 6, 96, pp. 481-493, 1989.
%H James Propp, <a href="http://jamespropp.org/moves15.pdf">Games of No Strategy and Low-Grade Combinatorics</a> (2015)
%e For n=4, 00400 relaxes to 01210 which relaxes to 02020. At this point, there are two ways to finish the game, so a(4)=2.
%t relax[L_] :=
%t relax[L] =
%t If[Min[L] < 0, 0,
%t If[Max[L] <= 1, 1,
%t Sum[relax[
%t Table[If[i == k - 1 || i == k + 1, L[[i]] + 1,
%t If[i == k, L[[i]] - 2, L[[i]]]], {i, 1, Length[L]}]], {k, 2,
%t Length[L] - 1}]]]
%t a[n_] := relax[
%t Join[Append[Table[0, {Floor[n/2]}], n], Table[0, {Floor[n/2]}]]]
%K nonn
%O 1,4
%A _James Propp_, Jul 30 2015