The juggler map is defined as \(J(n) = \lfloor n^{1/2} floor\) for even \(n\) and \(J(n) = \lfloor n^{3/2} floor\) 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 Collatz conjecture in its definition.