[seqfan] Constant with 2-Adic Valuation Continued Fraction

[Paul D. Hanna]

Thanks for the comments, Ed and Dave.

Quickly, I would like to compare the convergents of x and 2*x.
 
Notice how that the numerators and denominators of the 
convergents of x and 2*x are only different by a factor of 2:
 
Numerators to convergents of x:
1,3,4,19,23,65,88,769,857,2483,3340,15843,19183,54209,73392,...
Denominators to convergents of x:
1,2,3,14,17,48,65,568,633,1834,2467,11702,14169,40040,54209,...
 
Numerators to convergents of 2*x:
2,3,8,19,46,65,176,769,1714,2483,6680,15843,38366,54209,146784,...
Denominators to convergents of 2*x:
1,1,3, 7,17,24, 65,284, 633, 917,2467, 5851,14169,20020, 54209,...
 
This similarity is non-trivial given that the continued fraction
of 2*x has all those 2's inserted in the partial quotients. 
  
 
Regarding x^2, I am surprised that the continued fraction of x^2 
has doubly-exponential partial quotients, and so soon, 
considering that other powers of x seem to have relatively small 
partial quotients.  
   
Thanks, 
    Paul


On Wed, 17 Nov 2004 16:55:47 -0800 David C Terr
<David_C_Terr at raytheon.com> writes:

That's neat! I've gotten very interested in continued fractions myself
lately. I'm not too surprised that the coefficients of x^2 grow
exponentially, though I'm a bit surprised by how fast they grow. I think
it's true that if t has irrationality measure greater than 2, meaning
that its cfrac coefficients grow like e^e^kt for some positive constant
k, then those of t^2 do as well. I've done some numerical experiments on
such lines and it looks like it's true for every case I've tested. Please
correct me if I'm wrong though. Although x below has irrationality
measure 2, it is clearly not GK-regular (see def. in MathWorld), though
its coefficients have an obvious pattern. 

Dave 

"Paul D. Hanna" <pauldhanna at juno.com> 
11/17/2004 01:48 PM 
        
        To:        seqfan at ext.jussieu.fr 
        cc:         
        Subject:        [seqfan] Constant with 2-Adic Valuation Continued
Fraction

Consider the constant (newly added to OEIS as A100338):

x=1.353871128429882374388894084016608124227333416812118556923672649787...

The continued fraction of this constant is A006519 (greatest power of 2
dividing n): 
contfrac(x) = [1;2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,...A006519(n),... ] 
 
This constant x has the special property that the 
continued fraction expansion of 2*x is equal to the 
continued fraction expansion of x interleaved with 2's: 
contfrac(2*x) = [2;1, 2,2, 2,1, 2,4, 2,1, 2,2, 2,1, 2,8,...
2,A006519(n),...]. 

PARI code to get 1000 digits: 
\p1000
CF=vector(1500,n,2^valuation(n,2));
PQ=contfracpnqn(CF);
x=PQ[1,1]/PQ[2,1]*1.0

The continued fraction of x^2 is interesting: contfrac(x^2) = 
[1,1,4, 1,74, 1,8457, 1,186282390, 1,1,1,2,1,430917181166219,
11,37,1,4,2, 
41151315877490090952542206046, 11,5,3,12,2,34,2,9,8,1,1,2,7, 
13991468824374967392702752173757116934238293984253807017, ...] 
and some of the partial quotients of x^2 seem to grow exponentially. 
  
Has anyone seen this constant before? 
I wonder if it has some nice series representations as well ... 

Thanks,
   Paul 
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