Non-Trivial Induction Order

One of the standard proofs for the Arithmetic Mean-Geometric Mean Inequality is via the following induction route:

Let $n$ be the number of (non-negative) variables. It is proved for $n=1,2$ (straightforward), and then it is proved that if it is valid for $n \ge 2$ then it is valid for $2n$ and $n-1$.

This implies that the inequality is valid for all $n$ which are powers of 2, and since any positive integer is less than some power of 2, the inequality follows for all $n \ge 1$.

Are the more examples of this kind of induction (backward induction or $n \to 2n$), or in general not the usual $n \to n+1$ or $1,2,\cdots,n \to n+1$ induction schemes?

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This isn't too far from the usual scheme, but if you want to prove that every amount exceeding 7 cents can be made from 3-cent and 5-cent stamps, you can establish base cases 8, 9, and 10, and then prove that if you can do $n$, you can do $n+3$ by just adding one more 3-cent stamp.
Similarly you can show 6 divides $n^3-n$ by doing base cases 1, 2, ..., 6 and then doing $n\to n+6$.