For sufficiently large period lengths, the fraction of 1's in the repeating part tends to log(4/3)/log(2) = 0.415... as from the Gauss-Kuzmin distribution, i.e., a(n) tends to 0.415...*A013943(n) for sufficiently large A013943(n). - A.H.M. Smeets, Jun 02 2018

The "n-th nonsquare integer" in the definition is A005117(n + 1). - Michael B. Porter, Jun 06 2018