OEIS A000204: Lucas numbers

In this episode we dive into OEIS A000204, the Lucas numbers, starting with L1 = 1 and L2 = 3 and obeying L_n = L_{n-1} + L_{n-2}. They’re Fibonacci’s close cousins, sharing a recurrence but with a different seed. We explore their rich connections: identities with Fibonacci numbers (for example, L_n = F_{n-1} + F_{n+1}), and the closed form L_n = φ^n + ψ^n with φ = (1+√5)/2 and ψ = (1−√5)/2. We’ll see why L_n is the nearest integer to φ^n and how generating functions tie them to Fibonacci. The Lucas numbers pop up in concrete counting problems—number of matchings in an n-cycle, and the number of circular binary strings of length n with no consecutive 1s—plus tiling and ring-lattice interpretations. We’ll also touch on congruences: if p is prime, L_p ≡ 1 mod p, and L_{2^k} ≡ −1 mod 2^{k+1}, along with the notion of Lucas primes (often taken with the 2,1 seed). Finally, we’ll note how L_n relates to dynamical systems like the golden mean shift and to other OEIS entries, illustrating the deep web of connections tucked inside a deceptively simple recurrence.

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