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A350155 a(n) is the greatest number of times that a laser can hit a reflector on an n X n grid (see Comments for precise definition).

%I A350155 #41 Oct 10 2022 07:56:43
%S A350155 1,6,18,49,120,233
%N A350155 a(n) is the greatest number of times that a laser can hit a reflector on an n X n grid (see Comments for precise definition).
%C A350155 The problem is the following.
%C A350155 You are given an empty n X n grid. You can place some reflectors into the cells of the grid.
%C A350155 This arrangement consists of thin flat plates, each of which is rotatably mounted about a vertical axis so that two angular positions relative to the grid axes, namely plus and minus 45 degrees, are possible. A particle is shot into this arrangement and when it hits a reflector, it is deflected by 90 degrees without any loss of momentum according to the reflector's current orientation. After a reflector has been hit by the particle, it rotates by 90 degrees and maintains this position until the next hit. Not every grid position has to be assigned a reflector -- grid positions without reflectors are permitted.
%H A350155 Benjamin Butin, <a href="/A350155/a350155_1.txt">a(3) = 18</a>
%H A350155 Benjamin Butin, <a href="/A350155/a350155_2.txt">Exponential lower bound for A350155</a>
%H A350155 Dmitry Kamenetsky, <a href="https://puzzling.stackexchange.com/questions/112845/laser-and-mirrors-on-a-4x4-grid">Laser and mirrors on a 4x4 grid</a>, Puzzling StackExchange, November 2021.
%H A350155 Dmitry Kamenetsky and Daniel Mathias, <a href="/A350155/a350155_3.txt">Optimal solutions for n <= 6</a>
%H A350155 Dmitry Kamenetsky and Benjamin Butin, <a href="/A350155/a350155_5.txt">Improved patterns</a>.
%H A350155 Topcoder, <a href="https://www.topcoder.com/challenges/3a6b8bf2-1e02-4acb-a50a-09a13ed9ba34?tab=details">Marathon Match 132: BouncingBalls</a>.
%F A350155 a(n) >= 3*2^n-6, this value can be obtained with a simple pattern (see links above). - _Benjamin Butin_, Jan 20 2022
%e A350155 See links for examples.
%K A350155 nonn,more
%O A350155 1,2
%A A350155 _Dmitry Kamenetsky_, Dec 17 2021
%E A350155 a(3) corrected and a(4)-a(6) confirmed by _Benjamin Butin_, Jan 20 2022