[seqfan] Re: Special Continued Fraction Expansion

[Olivier Gerard]

On Wed, Jul 3, 2013 at 1:27 PM, Paul D Hanna <pauldhanna at juno.com> wrote:

> SeqFans,
>     Consider the constant(s) r such that the floor of the powers of r
> equals the partial quotients of the continued fraction of r:
>
> I wonder if it is easy to show that the following conjecture makes sense:
>
> Starting with a positive seed value r(o),
> if r(0) < sqrt(3), the recursion (**) restults in r1,
> if r(0) > sqrt(3), the recursion (**) restults in r2.
>
> The surprising thing (to me) is that the critical point would be r(0) =
> sqrt(3).
>
>
There is another critical point, namely 2  : the r1 results happen for
 sqrt(3) < r(0) < 2

There is the corresponding process with ceiling instead of floor, which is
more unstable
(many critical points between 1 and 5) again oscillating between two limit
values

r3 = 1.410988425021045794475591202911293....
r4 = 1.4344354347445048158696323513 ....

not far from  sqrt(2) with C.f.s:

1, 2, 2, 3, 4, 6, 8, 12, 16, 23, 32, 45, 63, 88, 124, 175, 247, 349, 492,
694, 979, 1381, 1948, 2749, 3878

1, 2, 3, 3, 5, 7, 9, 13, 18, 26, 37, 53, 76, 109, 157, 224, 322, 461, 662,
949, 1361, 1952, 2799, 4015, 5759


In general there are interesting things going on when you apply a constant
factor reasonably close and superior to 1

to the expression inside the floor or ceiling.