Odd period square roots

All square roots are periodic when written as continued fractions and can be written in the form:

\( \displaystyle \quad \quad \sqrt{N}=a_0+\frac 1 {a_1+\frac 1 {a_2+ \frac 1 {a3+ \dots}}} \)

For example, let us consider \( \sqrt{23} \):

\( \quad \quad \sqrt{23}=4+\sqrt{23}-4=4+\frac 1 {\frac 1 {\sqrt{23}-4}}=4+\frac 1 {1+\frac{\sqrt{23}-3}7} \)

If we continue we would get the following expansion:

\( \displaystyle \quad \quad \sqrt{23}=4+\frac 1 {1+\frac 1 {3+ \frac 1 {1+\frac 1 {8+ \dots}}}} \)

The process can be summarised as follows:

\[ \begin{align} &a_0=4, \frac 1 {\sqrt{23}-4}=\frac {\sqrt{23}+4} 7=1+\frac {\sqrt{23}-3} 7\\ &a_1=1, \frac 7 {\sqrt{23}-3}=\frac {7(\sqrt{23}+3)} {14}=3+\frac {\sqrt{23}-3} 2\\ &a_2=3, \frac 2 {\sqrt{23}-3}=\frac {2(\sqrt{23}+3)} {14}=1+\frac {\sqrt{23}-4} 7\\ &a_3=1, \frac 7 {\sqrt{23}-4}=\frac {7(\sqrt{23}+4)} 7=8+\sqrt{23}-4\\ &a_4=8, \frac 1 {\sqrt{23}-4}=\frac {\sqrt{23}+4} 7=1+\frac {\sqrt{23}-3} 7\\ &a_5=1, \frac 7 {\sqrt{23}-3}=\frac {7 (\sqrt{23}+3)} {14}=3+\frac {\sqrt{23}-3} 2\\ \\ &a_6=3, \frac 2 {\sqrt{23}-3}=\frac {2(\sqrt{23}+3)} {14}=1+\frac {\sqrt{23}-4} 7\\ &a_7=1, \frac 7 {\sqrt{23}-4}=\frac {7(\sqrt{23}+4)} {7}=8+\sqrt{23}-4\\ \end{align} \]

It can be seen that the sequence is repeating. For conciseness, we use the notation \( \sqrt{23}=[4;(1,3,1,8)] \), to indicate that the block \( (1,3,1,8) \) repeats indefinitely.

The first ten continued fraction representations of (irrational) square roots are:

\[ \begin{align} &\sqrt{2}=[1;(2)],\text{period=1}\\ &\sqrt{3}=[1;(1,2)],\text{period=2}\\ &\sqrt{5}=[2;(4)],\text{period=1}\\ &\sqrt{6}=[2;(2,4)],\text{period=2}\\ &\sqrt{7}=[2;(1,1,1,4)],\text{period=4}\\ &\sqrt{8}=[2;(1,4)],\text{period=2}\\ &\sqrt{10}=[3;(6)],\text{period=1}\\ &\sqrt{11}=[3;(3,6)],\text{period=2}\\ &\sqrt{12}=[3;(2,6)],\text{period=2}\\ &\sqrt{13}=[3;(1,1,1,1,6)],\text{period=5}\\ \end{align} \]

Exactly four continued fractions, for \( N \le 13 \), have an odd period.

How many continued fractions for \( N \le 10,000 \) have an odd period?