Two Families of Continued Fractions

Theorem 6   Let a_i = 2m(4m-1)^2n-i - 1, b_i = 8m(4m-1)^2n-i - 1, k = 4m(4m-1)²ⁿ - 1 and d = k² + 4(4m-1)²ⁿ, where $m, n \in \mathbb{Z}$ such that $m \geq 1$ and $n \geq 0$. Then,

$$\begin{multline*}\sqrt{d} = [k, \overline{a_{2n}, 1, b_1, a_{2n-1}, 1, b_2, \ld... ... 1, a_0, b_{2n}, 1, a_1, \ldots, b_1, 1, a_{2n}, 2k}] \mbox{.} \end{multline*}$$

  

Theorem 7   Let a_i and b_i be as they were in Theorem 1, let k = 4m(4m-1)²ⁿ⁺¹ - 1, and d = k² + 4(4m-1)²ⁿ⁺¹, where $m, n \in \mathbb{Z}$ such that $m \geq 1$ and $n \geq 0$. Then,

$$\begin{multline*}\sqrt{d} = [k, \overline{a_{2n}, 1, b_0, a_{2n-1}, 1, b_1, \ld... ... a_{-1}, b_{2n}, 1, a_0, \ldots, b_0, 1, a_{2n}, 2k}] \mbox{.} \end{multline*}$$