OEIS A000225: One-less-than-a-power-of-two (the all-ones binary numbers)

In this episode we explore A000225, the sequence 2^n − 1. Its binary form is n consecutive 1s, and it appears in many corners of combinatorics and CS. We’ll see how A000225 counts nonempty subsets of an n-element set, how A_{n+1} counts certain disjoint subset pairs (equivalently, pairs where at least one is empty), and how this ties to Gaussian binomial coefficients at q = 2. We’ll also connect to Stirling numbers, Pascal’s triangle, and the Tower of Hanoi minimum moves, as well as the length of the longest path in the n-dimensional hypercube. Finally, we’ll place A000225 in the broader family An = A^n − 1 for A ≥ 2 and discuss what these patterns reveal about how simple formulas reappear across mathematics.

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