cp's OEIS Frontend

A333329 Number of winnable configurations in Lights Out game (played on a digraph) summed over every labeled digraph on n nodes.

%I A333329 #27 Feb 16 2025 08:33:59
%S A333329 1,3,43,2619,654811,662827803,2699483026843,44102911693372059,
%T A333329 2886238576935227688091,756075355087132847491422363,
%U A333329 792522435884210281153847457333403,3323493099535510709729189614466101940379,55754039618636998102358059592995073452269940891
%N A333329 Number of winnable configurations in Lights Out game (played on a digraph) summed over every labeled digraph on n nodes.
%C A333329 Here a digraph may have at most one self loop (cf. A002416).  A winnable configuration is a subset of lit vertices that can be turned off by some toggling sequence.  In this version of the game, the digraph D is not necessarily symmetric so that the number of winnable configurations is 2^rank(A^t) where A^t is the transpose of the adjacency matrix of D.
%C A333329 In the limit as n goes to infinity, the probability that a random configuration on a random digraph is winnable is: Sum_{j>=0} (1/2^j) * (Product_{i>=j+1} (1-2^i))/(Product_{i>=1} (2^i - 2^(j-i))) = 0.610321...
%H A333329 A. Giffen and D. Parker, <a href="https://www.researchgate.net/publication/289241673_On_Generalizing_the_Lights_Out_Game_and_a_Generalization_of_Parity_Domination">On Generalizing the Lights Out Game and a Generalization of Parity Domination</a>, 2009.
%H A333329 L. Keough and D. Parker, <a href="https://arxiv.org/abs/1908.03649">An Extremal Problem for the Neighborhood Lights Out Game</a>, arXiv:1908.03649 [math.CO], 2019.
%H A333329 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LightsOutPuzzle.html">Lights Out Puzzle</a>
%F A333329 a(n) = Sum_{k=0..n} A286331(n,k)*2^k.
%F A333329 a(n) ~ c * 2^(n*(n+1)), where c = 0.610321518048266425924048782090628564983520109965690835927574616905934... - _Vaclav Kotesovec_, Apr 07 2020
%t A333329 Table[Table[2^k*Product[(2^n - 2^i)^2 /(2^k - 2^i), {i, 0, k - 1}], {k, 0, n}] // Total, {n, 0, 12}]
%K A333329 nonn
%O A333329 0,2
%A A333329 _Geoffrey Critzer_, Mar 15 2020