Interesting Mathematical Relationship

Today I found an interesting mathematical relationship where if you define f(x) such that

and you define f(0) = c and f(1) = 2c, then f(x) = 3f(x-1) for all x = 2.

Why is that?

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UPDATE: This was figured out in this Facebook thread. Also, it was found that the relationship is true for all x > 2 for f(0) and f(1) of any values.