  In our earlier publication we have shown how to compute by iteration a
rational number ${u_{2,k}}$ in the two-term Machin-like formula for pi of kind
$$\frac{\pi}{4}=2^{k-1}\arctan\left(\frac{1}{u_{1,k}}\right)+\arctan\left(\frac{1}{u_{2,k}}\right),\qquad
k\in \mathbb{Z},\quad k\ge 1,$$ where ${u_{1,k}}$ can be chosen as an integer
${u_{1,k}} = \left\lfloor{{a_k}/\sqrt{2-a_{k-1}}}\right\rfloor$ with nested
radicals defined as ${a_k}=\sqrt{2+a_{k-1}}$ and $a_0 = 0$. In this work we
report an alternative method for determination of the integer $u_{1,k}$. This
approach is based on a simple iteration and does not require any irrational
(surd) numbers from the set $\left\{a_k\right\}$ in computation of the integer
$u_{1,k}$. Mathematica programs validating these results are presented.
