Suppose r>0. Let's mimic the standard algorithm for finding the simple
continued fraction of r, but, instead of taking successive numerators
=1, take each to be the most recently generated "a(k)" - like this:
r(0)=r, a(0)=[r(0)]
r(1)=a(0)/(r(0)-a(0)) a(1)=[r(1)]
r(2)=a(1)/(r(1)-a(1)) a(2)=[r(2)], and so on.
Call the sequence <a(0),a(1),a(2),...> the self-generating continued
fraction of r, and denote it by S(r).
Has anyone encountered this previously?
One very attractive example has been known for a long time:
1/(e-2) = <1,2,3,4,5,6,7,8,...>.
An easy problem is to determine all r for which S(r) is a constant
sequence.
Probably more interesting, though, are examples like these:
<1,2,4,8,16,32,64,...> = 1.4086159797...
<1,3,5,7,9,11,13,...> = 1.2831923416...
<1,4,7,10,13,16,...> = 1.221107010123...
Are these numbers "known"?
Let R_n denote the set of r corresponding to the arithmetic sequences of
positive integers congruent to k mod n for k=1,2,...,n. What can be
said about the numbers in R_n?
Best wishes,
Clark Kimberling
