Juggler sequence

The juggler map is defined as \(J(n) = \lfloor a_n^{1/2} \rfloor\) for even \(n\) and \(J(n) = \lfloor a_n^{3/2} \rfloor\) for odd \(n\). A juggler sequence, then, is a type of positive integer sequence defined recursively by a given base value \(a_0\) and the recurrence relation \(a_{n + 1} = J(a_n)\). Juggler sequences were first defined by Clifford Pickover.

Juggler sequences tend to grow rapidly, decline, and stabilize at \(J(1) = 1\). It is an unproven conjecture that all juggler sequences end up stabilizing at 1; this has parallels to in its definition.

Certain initial values create juggler sequences that reach unusually high maxima. \(a_0 = 37\) creates a trajectory that peaks at 24906114455136, and \(a_0 = 30817\) peaks at a value with 45391 digits.