The Franel numbers A00172 are defined by A(n) = sum_{k=0}^n binom(n,k)^3. In this episode we explore their elegant second-order recurrence (n+1)^2 A(n+1) = (7n^2+7n+2) A(n) - 8 n^2 A(n-1), and their surprising appearances across combinatorics, number theory, and geometry—from counting problems and Nash equilibria to Apery-like phenomena and prime-modulus congruences.

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