Criteria for apwenian sequences

In 1998, Allouche, Peyrière, Wen and showed that the Hankel determinant Hn of Thue-Morse sequence over {?1,1} satisfies Hn/2n?1?1(mod2) for all n?1. Inspired by this result, Fu Han introduced apwenian sequences {?1,1}, namely, ±1 whose determinants satisfy n?1, proved with computer assistance a few are apwenian. paper, we obtain an easy to check criterion sequences, which allows us determine fixed points substitutions constant length. Let f(z) be generating functions such sequences. We show integer b?2 f(1/b)?0, real number f(1/b) is transcendental its irrationality exponent equal 2. Besides, also derive 0-1 Hn?1(mod2) find only sequence, among length, period-doubling sequence. Various examples given projection provided. Furthermore, prove Sturmian or {0,1} not And conjecture substitution non-constant length can