OEIS A000283: Sum of squares recurrence

We explore the simple rule A_n = A_{n-1}^2 + A_{n-2}^2 with A_0 = 0, A_1 = 1. The early terms are tiny (0, 1, 1, 2, 5, 29, 866, 750797, ...) but the sequence explodes incredibly fast, essentially doubling the exponent at each step. Yet there’s a striking continuous shadow: A_n ≈ floor( A^{2^{n-1}} ) with A ≈ 1.23539. We unpack how a humble recurrence links discrete growth to a continuous approximation, touch on Benford’s Law, and note the computational challenges and tools used to experiment with these gigantic integers.

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