Explore A000051, the sequence a(n) = 2^n + 1. We’ll trace its tidy binary shape (11, 101, 1001, …), its rare square at n = 3 (9) proven by de Bessy in the 1600s, and its surprising web of connections—from Fermat primes and prime curiosities to Hilbert curves, Pisano periods, and Cunningham numbers. A simple rule that opens a doorway to prime theory, fractals, and computer graphics.

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