OEIS A000123: Binary partitions

We explore A000123, the binary partition function: the number of ways to write n as a sum of powers of two with unlimited repetition. We unpack the recurrence A(n) = A(n−1) + A(⌊n/2⌋), the striking fact that every A(n) for n > 0 is even, and the elegant generating function ∏_{k≥0} 1/(1 − x^{2^k}). Along the way we connect to coin-change problems, Euler’s partitions, and the sequence’s rich history—from Churchhouse and Gupta to de Bruijn and Knuth—and discuss how its growth sits between polynomial and exponential, with generalizations and open questions to ponder.

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