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This is a fun example. The formula's easy to prove by induction, but discovering it is less obvious and this is an interesting way.

It's the tip of the iceberg though - generatingfunctionology is worth inspecting thoroughly.




There are actually perfectly routine ways to discover formulas like this one for this type of equation. See http://en.wikipedia.org/wiki/Recurrence_relation for some of the theory behind how to do it.




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