π in continued fractions

Since π is an irrational number, there's no way to express it exactly as a decimal or a simple fraction.  The next best thing is to approximate it.  The use of continued fractions is an elegant yet practical way to approximate π.  The simple continued fraction approximating π is:

which is conveniently represented by the notation [3;7,15,1,292,1,1,1,2,1,3,1,...].

By truncating this infinite continued fraction to finite expansions, we get the following fractional approximations:

[3] = 3
[3;7] = 22/7
[3;7,15] =333/106
[3;7,15,1] = 355/113
[3;7,15,1,292] = 103993/33102
etc.

Unfortunately, there are no regular patterns shown in the simple continued fraction for π.