Extraction problem of the Pell sequence (Q2707840)

Let \(\alpha\in [0,1]\) be an irrational number. The characteristic sequence of \(\alpha\) is an infinite binary sequence \(f\) whose \(n\)-th terms is \([(n+1)\alpha] -[n\alpha]\) for \(n\geq 1\). Let \(s_m\) denote the prefix of \(f\) of length \(m\) and let \(f_m\) denote the suffix of \(f\) with \(f=s_m f_m\) for \(m\geq 0\). Let \(f_0=f\) and let \(R\) be the reversion operator, that for an alphabet \(A\), if \(c_1,c_2, \dots, c_n\in A\) for \(n\geq 1\), then \(R(c_1c_2\dots c_n)=c_n \dots c_2c_1\).NEWLINENEWLINENEWLINEThe basic assertion is Theorem 2.1: (a) Let \(\alpha= \sqrt 2-1\) and let \(f\) be the characteristic sequence of \(\alpha\). Then \(\langle f,f_m \rangle=R(s_m)\) for all \(m\geq 1\). (b) If \(n\geq 1\), \(r_1,r_2, \dots\) is an infinite sequence of integers with \(0\leq r_1\leq 1\), \(0\leq r_i\leq 2\) (\(2\leq i\leq n)\), and \(r_i=0\) \((i>n)\), then NEWLINE\[NEWLINE\langle u_0u^2_1u^2_2 \dots,u_0^{1-r_1} u_1^{2-r_1} u_2^{2-r_1} \dots\rangle =u_0^{r_1} u_1^{r_1} \dots u_{n-1}^{r_1}.NEWLINE\]NEWLINE The paper is a continuation of the author's previous papers [ibid. 33, 113-122 (1995; Zbl 0842.11007), Discrete Math. 177, 33-50 (1997; Zbl 0890.68108)].