sequences of convergents of constant continued fractions

[Marc LeBrun]

Pebble in the Shoe Dept.:

This small item has been on my "good intentions list" for too long; lest I 
forget it, maybe someone else would like to help?

There is a certain little group of related sequences in the OEIS which I 
think ought to be linked together and their entries standardized.

These are the sequences where a(n+1)/a(n) is the nth convergent of x = 
/k,k,k,.../.

That is, whose successive ratios are the convergents of the special 
continued fractions wherein all of the CF terms are the same.

As you well know, the Fibonacci numbers are the case where k=1 and so x = 
tau (aka phi) the golden ratio.

In general, x = (k + sqrt(k^2 + 4))/2, and the sequence obeys the simple 
2-term recurrence a(n+1) = k a(n) + a(n-1).

I believe these sequences are particularly special because they are the 
only integer sequences where ALL term ratios are convergents.  This is 
because of the constraint that a(n>1) must appear as both a numerator and a 
denominator (originally I had assumed this class to be much larger).  As a 
result, unlike most CF expansions, the single sequence gives both the 
numerators and denominators.

Happily the OEIS currently contains entries for several (a dozen?) values of k.

But it would be good to fill in any missing k (up to k=12 or 20 say?) and 
then collect all of them together in a coherently cross-linked group.

Also, comments should be added to explicitly give the values of k, x and/or 
the recurrence (sometimes the submitter may not have been aware of all 
three, or describes them opaquely).

Finally, the entries should be "standardized" with regard to the starting 
index of n, the description of x (often the fraction x-k is cited instead) 
and similar such issues.  In some cases this may require submitting a whole 
new trivially different entry.

To do this someone with suitable software (or stamina!<;-) will have to 
lookup all the existing entries for some range of k of interest, and add 
the necessary comments and revisions, as well as any new sequences.

If anyone would like to take this on, it would be a nice little enhancement.

Thanks!