On the approximation by three consecutive continued fraction convergents

Denote by pn/qn,n=1,2,3,…pn/qn,n=1,2,3,…, the sequence of continued fraction convergents of the real irrational number xx. Define the sequence of approximation coefficients by θn:=qn|qnx−pn|,n=1,2,3,…θn:=qn|qnx−pn|,n=1,2,3,…. A laborious way of determining the mean value of the sequence |θn+1−θn−1|,n=2,3,…|θn+1−θn−1|,n=2,3,…, is simplified. The method involved also serves for showing that for almost all xx the pattern θn−1<θn<θn+1θn−1<θn<θn+1 occurs with the same asymptotic frequency as the pattern θn+1<θn<θn−1θn+1<θn<θn−1, namely 0.12109⋯0.12109⋯. All the four other patterns have the same asymptotic frequency 0.18945⋯0.18945⋯. The constants are explicitly given.