OEIS A000302: Powers of Four

In this Deep Dive we explore A000302 — the powers of 4 — a deceptively simple sequence that secretly ties together number theory, combinatorics, and beyond. We start from 4^n and observe every term is a perfect square, linking powers of four to powers of two. But the surprises don’t stop there: A000302 is described as the convolution square root of the central binomial coefficients A000984, revealing hidden combinatorial structure beneath the surface. We dive into interpretations like compositions of n with parts less than four, four-colored compositions with no adjacent equal colors, and connections to Dick paths via peak heights. We also see how it pops up in Benford’s law and in counting nodes at level n in a balanced four-ary tree. The takeaway: simple exponential sequences can serve as foundational bridges across diverse areas of math, teaching us to spot patterns and the unity underlying complex systems.

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