"Egyptian Continued Fraction" Expansions

[Leroy Quet]

>     The recent discussion of Egyptian fractions reminded me of something
>that Hans Havermann and I worked on a while back.
>I call these expansions, "Egyptian Continued Fractions" (if I may).
>
>The "Egyptian continued fraction" (ECF) expansion of x is still a series
>using a greedy algorithm, but instead of unit fractions, simple continued
>fractions are used.  The smallest partial quotients {a(n)} are chosen so
>that the following sum of continued fractions approach, but do not
>exceed, x:
>  x = a(0) + [0;a(1)] + [0;a(2),a(1)] + [0;a(3),a(2),a(1)] + ... 
>
>....
 [lots snipped]


Your idea reminds me of a post a made a few years ago to sci.math.

Even though my idea is different, in that I have the number of terms in 
each continued-fraction remain constant (or under a fixed m), it is 
connected immediately with this discussion.

I wrote, in "Continued Egyptian Fractions":

Consider a generalization of Egyptian fractions, where positive
rationals are constructed by the finite sum of the m_th-order
"Egyptian fractions",
[0;a[1], a[2], a[3],..., a[m]]. 
(The "[...]" is a continued fraction, and the a's are positive
integers.)

Of course, the standard Egyptian fractions are the first-order
generalized Egyptian fractions.

Note: Since 1/1 is considered an acceptable standard Egyptian fraction
(I think), then perhaps a[m] should be allowed to be 1.

We might consider the sum of, say order-2 Egyptian fractions.
Example:
Since 1/(1 +1/2) = 2/3, 1/(2 +1/1) = 1/3, 1/(2 +1/3) = 3/7;
then 10/7 can be constructed from order-2 Egyptian fractions as
10/7 = 2/3 + 1/3 + 3/7 .

I guess the question is, given an m, which positive rationals are the
sums of a finite number of m-order Egyptian fractions?

Thanks,
Leroy Quet