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A165738 Rank deficiency (= dimension of the null space) of the n X n "Lights Out" puzzle on a torus.

%I A165738 #31 Feb 16 2025 08:33:11
%S A165738 0,0,4,0,8,8,0,0,4,16,0,16,0,0,12,0,16,8,0,32,4,0,0,32,8,0,4,0,0,24,
%T A165738 40,0,44,32,8,16,0,0,4,64,0,8,0,0,12,0,0,64,0,16,20,0,0,8,8,0,4,0,0,
%U A165738 48,0,80,52,0,56,88,0,64,4,16,0,32,0,0,12,0,0,8,0,128,4,0,0,16,24,0,4,0,0,24
%N A165738 Rank deficiency (= dimension of the null space) of the n X n "Lights Out" puzzle on a torus.
%C A165738 The number of solutions to the puzzle is 2^a(n). If a(n)=0 then the puzzle has a unique solution.
%D A165738 See A075462 for further references.
%H A165738 Max Alekseyev and Thomas Buchholz, <a href="/A165738/b165738.txt">Table of n, a(n) for n = 1..1000</a>. (Terms n=91 through n=1000 from Thomas Buchholz, May 20 2014)
%H A165738 M. Anderson and T. Feil, <a href="http://www.jstor.org/stable/2690705">Turning Lights Out with Linear Algebra</a>, Mathematics Magazine, 71 (1998), 300-303.
%H A165738 Andries E. Brouwer, <a href="http://www.win.tue.nl/~aeb/ca/madness/madrect.html">Lights Out and Button Madness Games</a>. [Gives theory and a(n) for n = 1..1000, Jun 19 2008]
%H A165738 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LightsOutPuzzle.html">Lights Out Puzzle</a>
%F A165738 a(n) <= 2n.
%F A165738 a(n) is a multiple of 4 and satisfies a(2n) = 2a(n). a(n+1) = 2 * A159257(n) + 4 if n = 2 (mod 3) and a(n+1) = 2 * A159257(n) otherwise. - _Thomas Buchholz_, May 22 2014
%Y A165738 Cf. A159257.
%K A165738 nonn
%O A165738 1,3
%A A165738 _Max Alekseyev_, Sep 25 2009
%E A165738 More terms from _Thomas Buchholz_, May 20 2014