Investigating Small Sequences

Now let us have a preliminary look at the differences between the radix and continued fraction representations. The function allNDigitSequences returns a list of all -digit base sequences. We will always assume that the first digit is zero.

The function sequencesAnds adds the difference between the radix and continued fraction representations to the sequences.

Here we show the differences for the simple and nonsimple continued fractions for sequences of length 2 to 5. We color the differences according to the original sequence length.

The second graphic shows the partially periodic, fractal-like behavior of the differences. For these short sequences, we can obtain a few coinciding digits for 0.1... and 0.3... for simple continued fractions. The next graphics zoom into these areas.

Here we show base 2 using longer sequences, all sequences up to length 12. Qualitatively we obtain similar graphics.