Decimal expansion of the square root of a nonnegative integer

Abstract

For positive integers $n$ and $k$, with $k \leq 2n$, let $\sqrt{n^2+k} = n_t \cdots n_1.a_1a_2a_3 \cdots$
be the decimal expansion of $\sqrt{n^2+k}$. In this paper, we introduce a systematic
method of how to calculate the value of $a_i$ for all $i =1, 2, \dots$.