OEIS A000278: Quadratic Fibonacci recurrence

We explore A000278, defined by a(n) = a(n-1) + [a(n-2)]^2 with initial values a(0)=0, a(1)=1. The sequence starts 0, 1, 1, 2, 3, 7, 16, 65, and its nonlinear square term leads to explosive, doubly exponential growth. Remarkably, the growth splits by parity: the even- and odd-index subsequences follow two distinct asymptotic tracks, with constants around 1.66875 and 1.69306, respectively. Beyond growth, A000278 counts certain combinatorial structures—forests built under recursive rules—giving a concrete interpretation to these colossal numbers. This is a striking example of how a simple nonlinear recurrence can encode rich structure and reveal subtle long-term behavior.

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