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A333329 Number of winnable configurations in Lights Out game (played on a digraph) summed over every labeled digraph on n nodes. 0
1, 3, 43, 2619, 654811, 662827803, 2699483026843, 44102911693372059, 2886238576935227688091, 756075355087132847491422363, 792522435884210281153847457333403, 3323493099535510709729189614466101940379, 55754039618636998102358059592995073452269940891 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

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...

LINKS

Table of n, a(n) for n=0..12.

A. Giffen and D. Parker, On Generalizing the Lights Out Game and a Generalization of Parity Domination, 2009.

L. Keough and D. Parker, An Extremal Problem for the Neighborhood Lights Out Game, arXiv:1908.03649 [math.CO], 2019.

Eric Weisstein's World of Mathematics, Lights Out Puzzle

FORMULA

a(n) = Sum_{k=0..n} A286331(n,k)*2^k.

a(n) ~ c * 2^(n*(n+1)), where c = 0.610321518048266425924048782090628564983520109965690835927574616905934... - Vaclav Kotesovec, Apr 07 2020

MATHEMATICA

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}]

CROSSREFS

Sequence in context: A307248 A009720 A300873 * A201173 A290777 A309401

Adjacent sequences:  A333326 A333327 A333328 * A333330 A333331 A333332

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Mar 15 2020

STATUS

approved

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Last modified April 19 20:35 EDT 2021. Contains 343117 sequences. (Running on oeis4.)