OEIS A000304: A multiplicative recurrence tied to Fibonacci exponents

We examine OEIS sequence A000304, defined by a_n = a_{n-1} a_{n-2} with starting values a_2 = 2 and a_3 = 3 (offset 2). The terms grow unbelievably fast: 2, 3, 6, 18, 108, 1944, 209952, ... Yet there is a clean explicit form: for n ≥ 3, a_n = 2^{F_{n-3}} 3^{F_{n-2}}, where F_k are the Fibonacci numbers (F_0 = 0, F_1 = 1). We’ll unpack why the simple multiplicative rule translates into Fibonacci exponents, verify with small n (n = 4, 5, 6), and discuss how the OEIS page connects A000304 to Fibonacci and related sequences. A compact example of how additive sequences (Fibonacci) govern a multiplicative recurrence in number theory.

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