Solution

Find the first several terms of the sequence associated with the simple continued fraction representation for $\pi$.


There is no apparent pattern to the simple continued fraction representation for $\pi$, as can be seen below:

$$\pi = [3; 7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,\ldots]$$

That said, there are some beautiful non-simple continued fraction representations for $\pi$, however:

$$\pi=\textstyle \frac{4}{1+\textstyle \frac{1^2}{2+\textstyle \frac{3^2}{2+\textstyle \frac{5^2}{2+\textstyle \frac{7^2}{2+\textstyle \frac{9^2}{2+\ddots}}}}}} =3+\textstyle \frac{1^2}{6+\textstyle \frac{3^2}{6+\textstyle \frac{5^2}{6+\textstyle \frac{7^2}{6+\textstyle \frac{9^2}{6+\ddots}}}}} =\textstyle \frac{4}{1+\textstyle \frac{1^2}{3+\textstyle \frac{2^2}{5+\textstyle \frac{3^2}{7+\textstyle \frac{4^2}{9+\ddots}}}}}$$