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).

1, 6, 18, 49, 120, 233

Offset: 1

Author

Dmitry Kamenetsky, Dec 17 2021

Keywords

• nonn
• more

Comments

The problem is the following.

You are given an empty n X n grid. You can place some reflectors into the cells of the grid.

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.

Examples

			See links for examples.
		

Links

• Benjamin Butin, a(3) = 18
• Benjamin Butin, Exponential lower bound for A350155
• Dmitry Kamenetsky, Laser and mirrors on a 4x4 grid, Puzzling StackExchange, November 2021.
• Dmitry Kamenetsky and Daniel Mathias, Optimal solutions for n <= 6
• Dmitry Kamenetsky and Benjamin Butin, Improved patterns.
• Topcoder, Marathon Match 132: BouncingBalls.

Formula

a(n) >= 3*2^n-6, this value can be obtained with a simple pattern (see links above). - Benjamin Butin, Jan 20 2022

Extensions

a(3) corrected and a(4)-a(6) confirmed by Benjamin Butin, Jan 20 2022