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Stacks of Reflecting Plates

A ray of light enters a stack of glass plates. The ray can either pass through a plate or be reflected by it. This Demonstration shows all the possible ways the ray has to leave the stack after reflections. If is even, the ray traverses the stack, otherwise it leaves the stack on the same side it entered.
With plates, the surfaces are labeled , with denoting the lower surface of the lowest plate, and a ray with reflections is uniquely determined by surface labels.

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With two plates the number of paths with reflections is the Fibonacci number . With three plates the number of paths is for , which is Sloane's sequence A006356.
References:
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, 2nd ed., Reading, MA: Addison-Wesley, 1994 p. 291.
Sequence A006356 in N. J. A. Sloane, ed., The On-Line Encyclopedia of Integer Sequences, 2008.
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