Output the Juggler sequence
Julia, 64 50 48 42 32 30 bytes



g(x)=[x;x&lt;3||g(x^(x%2+.5)÷1)]


This is a recursive function that accepts an integer and returns a float array.

We build an array by concatenating the input with the next term of the sequence, computed as x to the power of its parity plus 1/2. This gives us either x1/2 or x1+1/2 = x3/2. Integer division by 1 gets the floor. When the condition x &lt; 3 is true, the final element will be a Boolean rather than a numeric value, but since the array is not of type Any, this is cast to have the same type as the rest of the array.

Saved 14 bytes thanks to Dennis!