The series representation \[ x= \frac{1}{d_1}+ \frac{a_1(d_1)} {b_1(d_1)} \frac{1}{d_2}+ \frac{a_1(d_1)\cdots a_n(d_n)} {b_1(d_1)\cdots b_n(d_n)} \frac{1}{d_{n+1}}+\cdots, \] \(a_n(j)\), \(b_n(j)\) \((n\geq 1)\) are positive integer valued functions of the integers \(j\geq 1\), \(d_n\) are positive integers, \(h_n(j)= \frac{a_n(j)} {b_n(j)} j(j-1)\), \(d_{n+1}> h(d_n)\). This yields \(d_n\geq 2\) \((n= 1,2,\dots)\). A modification of the algorithm leading to this series representation gives an alternating representation \[ x= \frac{1}{D_1}- \frac{a_1(D_1)} {b_1(D_1)} \frac{1}{D_2}+\cdots+ (-1)^n \frac{a_1(D_1)\cdots a_n(D_n)} {b_1(D_1)\cdots b_n(D_n)} \frac{1}{D_{n+1}}+\cdots\;. \] The paper is devoted to the study of the sequence \(D_n(x)\) and related sequences from the metric point of view.