  Automatic sequences such as the Thue-Morse sequence and the Rudin-Shapiro
sequence are highly predictable and thus not suitable in cryptography. In
particular, they have small expansion complexity. However, they still have a
large maximum order complexity. Certain subsequences of automatic sequences are
not automatic anymore and may be attractive candidates for applications in
cryptography. In this paper we show that subsequences along the squares of
certain pattern sequences including the Thue-Morse sequence and the
Rudin-Shapiro sequence have also large maximum order complexity but do not
suffer a small expansion complexity anymore.
