Equivalence of Two Fibonacci Number Interpretations

This Demonstration illustrates the equivalence of these two interpretations of the Fibonacci numbers :
• the number of ways to tile a checkerboard with dominoes
• the number of ways a ray of light can pass through two panes of glass with reflections
In the figure, when viewing reflections, the top row of the checkerboard pattern represents one pane of glass, and the bottom row represents the second pane of glass; reflections can occur on any of the three "surfaces" of the panes.

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DETAILS

References:
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, 2nd ed., Reading, Massachusetts: Addison-Wesley, 1994.
M. Gardner, Mathematical Circus, New York: Alfred A. Knopf, 1979.
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Related Curriculum Standards

US Common Core State Standards, Mathematics