I have been working on an article at
https://oeis.org/wiki/Table_of_convergents_constants

where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) for a few numbers.

It would be nice to support the article with some quality analysis.

Before June 9, 2011, was starting to extract and clearly define a pattern to these constants cf the article. I've made some progress in finding the patterns to their continued fractions. Can you find the next pattern $a_6(n)$? If it is like $a_5(n)$ it will be dependent on the moduli of some natural number $K$.

Beginning with $n=0$, the sequence is:

Edit [I made this to hard with an error. I should have said,
Beginning with $n=1$, the sequence is:]

1, 2, 1, 3, 3, 9, 4, 1, 5, 2, 7, 9, 8, 1, 9, 2, 11, 2, 12, 1, 13, 2, 15, 16, 16, 1, 17, 2, 19, 1, 20, 1, 21, 2, 23, 1, 24, 1, 25, 2, 27, 1, 28, 1, 29, 2, 31, 1, 32, 1, 33, 2, 35, 2, 36, 1, 37, 2, 39, 2, 40, 1, 41, 2, 43, 2, 44, 1, 45, 2, 47, 2, 48, 1, 49, 2, 51, 3, 52, 1, 53, 2, 55, 3, 56, 1, 57, 2, 59, 3, 60, 1, 61, 2, 63, 3, 64, 1, 65, 2, 67, 4, 68 ,1, 69, 2, 71, 4, 72, 1, 73, 2, 75, 4, 76, 1, 77, 2, 79, 4, 80, 1, 81, 2, 83, 4, 84, 1, 85, 2, 87, 5, 88, 1, 89, 2, 91, 5, 92, 1, 93, 2, 95, 5, 96, 1, 97, 2, 99, 5, 100, 1, 101, 2, 103, 6, 104, 1, 105, 2, 107, 6, 108, 1, 109, 2, 111, 6, 112, 1, 113, 2, 115, 6, 116, 1, 117, 2, 119, 7, 120, 1, 121, 2, 123, 7, 124, 1, 125, 2, 127, 7, 128, 1, 129, 2, 131, 7, 132, 1.

The function for the pattern to the sequence could be piecewise defined, as the last known piece of $a_5(n)$ started at $n=24$. Whether the functions for $a_5(n)$ and $a_6(n)$ can be defined without piecewise functions has yet to be answered as far as I know.

Addendum June 10, 2011 [Now that I have $a_1(n)$ through $a_6(n)$, I find it difficult to find what they all have in common. Perhaps there is an easier pattern to find $a_1(n)$ from $a_0(n)$, $a_2(n)$ from $a_1(n)$, ...? Can you help me find it? I will ask something like this in the talk page soon.]