Define the *three* sequences,

The last two are *Apery numbers *and have been discussed previously. The first are the *central Delannoy numbers *which obeys the limit,

which is the square of the silver ratio. (The ratios for the others have already been mentioned.) These have the *recurrence relations*,

To recall, the polynomials and generated numbers for the continued fractions of , so I was curious if could be used in an analogous manner. It turns out, depending on what sign to use, it gives either or ,

or,

where, starting with *n* = 1,

This can be partly demystified since one continued fraction for the natural logarithm, and arctan, is,

and,

Hence, if , as with the case {*x, y, z*} = {1, 2, 3}, then there will be identical-looking continued fractions that differ only in the signs. But it remains interesting how the recurrence relations of these three binomial sums are involved in the continued fractions of . Later, we shall see there is a recurrence relation for the cfrac of as well.

### Like this:

Like Loading...

*Related*