Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Consider the sequence

a₁=1, a_n+1=[√(2a_n(a_n+1))] for n≥1 where [x] is the floor function.

Here are the first 17 terms and the alternate differences a_2k+1 - a_2k

1 2 3 4 6 9 13 19 27 38 54 77 109 154 218 309 437
   1   2   4     8    16    32      64      128

Are they all powers of 2?