OEIS A000308: Recursive Product Growth

We dive into A000308, defined by a(n) = a(n-1) · a(n-2) · a(n-3) with starting values 1, 2, 3. The growth is astonishingly fast: 1, 2, 3, 6, 36, 648, 139,968, …, a true example of hyper-exponential behavior from a simple triple-product rule. With non-negative initials the sequence stays positive and climbs without bound; zeros or negatives would alter the behavior dramatically. The OEIS entry also presents an explicit formula tying A000308 to powers of 2 and 3 whose exponents are governed by Tribonacci-type sequences (A000073 and A001590), revealing a surprising link between additive recurrences (like Tribonacci) and this multiplicative one. It’s a striking illustration of how different recursive patterns can connect, offering insights for number theory students into the hidden structure behind explosive growth and the practical limits it would impose in real-world scaling scenarios.

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