A Continued Fraction Derived From Recursion

Leroy Quet qqquet at mindspring.com
Sun Mar 23 00:23:27 CET 2003

Posted this to sci.math:

Consider this sequence from the On-Line Encyclopedia Of Integer Sequences:

>ID Number: A053978
>Sequence:  1,1,2,2,8,59,1,46,1,2,3,22,1,60,1,1,1,4,1,6,2,1,1,3,4,1,2,6,
>           1,25,2,1,2,7,3,11,1,1,20,1,3,1,1,3,1,11,1,2,31,3,2,2,5,1,1,
>           3,3,11,1,2,4,2,1,1,1,6,3,3,3,15,2,1,5,2,2,3,2,1,1,3,2,2,8,1,
>           7,1,4,1,4,2,2,5,2,1,4,1,19,1
>Name:      Continued fraction expansion of limit_{m->infinity} [r_m],
>              where r_1=1, r_{m+1}= r_1 +1/(r_2 +1/(r_3 +...1/(r_{m-1}
>              +1/r_m)...)).
>Comments:  limit_{m->infinity} [r_m] =1.7118691868...
>Keywords:  cofr,nice,nonn
>Offset:    0
>Author(s): Leroy Quet (qqquet at mindspring.com), Apr 02 2000
>Extension: More terms from Jeppe Stig Nielsen (sequence at jeppesn.dk), Apr 
>13 2000

[edited so that "{r->infinity}" in 'Comments-line' is replaced with 


does the limit, as m -> oo, of the geometric-average of the first m terms 
of this sequence approach Khinchtine's constant??



If r_1 = x = any positive real,

and we let limit(m->oo} r_m = R(x),

then R(x) has a minimum at some particular x between 0 and 1.

Let this x be X.

And let Y = R(X).

What are the simple(ie.with positive integer terms) continued-fraction 
representations of X and Y?

And, the obvious question, do the geometric-averages of X's and Y's 
simple CF terms approach Khinchtine's constant??

Leroy Quet

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