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<DIV>Thanks for the comments, Ed and Dave.</DIV>
<DIV> </DIV>
<DIV>Quickly, I would like to compare the convergents of x and
2*x.<BR> <BR>Notice how that the numerators and denominators of the
<BR>convergents of x and 2*x are only different by a factor of
2:<BR> <BR>Numerators to convergents of
x:<BR>1,3,4,19,23,65,88,769,857,2483,3340,15843,19183,54209,73392,...<BR>Denominators
to convergents of
x:<BR>1,2,3,14,17,48,65,568,633,1834,2467,11702,14169,40040,54209,...<BR> <BR>Numerators
to convergents of
2*x:<BR>2,3,8,19,46,65,176,769,1714,2483,6680,15843,38366,54209,146784,...<BR>Denominators
to convergents of 2*x:<BR>1,1,3, 7,17,24, 65,284, 633, 917,2467,
5851,14169,20020, 54209,...<BR> <BR>This similarity is non-trivial given
that the continued fraction<BR>of 2*x has all those 2's inserted in the partial
quotients. </DIV>
<DIV> <BR> <BR>Regarding x^2, I am surprised that the continued
fraction of x^2 <BR>has doubly-exponential partial quotients, and so soon,
<BR>considering that other powers of x seem to have relatively small <BR>partial
quotients. </DIV>
<DIV> <BR>Thanks, <BR> Paul</DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>On Wed, 17 Nov 2004 16:55:47 -0800 David C Terr <<A
href="mailto:David_C_Terr@raytheon.com">David_C_Terr@raytheon.com</A>>
writes:</DIV>
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style="PADDING-LEFT: 10px; MARGIN-LEFT: 10px; BORDER-LEFT: #000000 2px solid">
<DIV><BR><FONT face=sans-serif size=2>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.</FONT> <BR><BR><FONT face=sans-serif
size=2>Dave</FONT> <BR></DIV>
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<TD><FONT face=sans-serif size=1><B>"Paul D. Hanna"
<pauldhanna@juno.com></B></FONT>
<P><FONT face=sans-serif size=1>11/17/2004 01:48 PM</FONT> <BR></P>
<TD><FONT face=Arial size=1> </FONT><BR><FONT
face=sans-serif size=1> To:
seqfan@ext.jussieu.fr</FONT> <BR><FONT face=sans-serif
size=1> cc:
</FONT> <BR><FONT face=sans-serif size=1>
Subject: [seqfan] Constant with 2-Adic
Valuation Continued Fraction</FONT></TR></TBODY></TABLE><BR><FONT
face="Times New Roman" size=3>Consider the constant (newly added to OEIS as
A100338):<BR><BR>x=1.353871128429882374388894084016608124227333416812118556923672649787...<BR></FONT><BR><FONT
face="Times New Roman" size=3>The continued fraction of this constant is
A006519 (greatest power of 2 dividing n):</FONT> <BR><FONT
face="Times New Roman" size=3>contfrac(x) =
[1;2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,...A006519(n),... ] </FONT><BR><FONT
face="Times New Roman" size=3> <BR>This constant x has the special
property that the <BR>continued fraction expansion of 2*x is equal to the
<BR>continued fraction expansion of x interleaved with 2's: <BR>contfrac(2*x)
= [2;1, 2,2, 2,1, 2,4, 2,1, 2,2, 2,1, 2,8,... 2,A006519(n),...]. <BR><BR>PARI
code to get 1000 digits:
<BR>\p1000<BR>CF=vector(1500,n,2^valuation(n,2));<BR>PQ=contfracpnqn(CF);<BR>x=PQ[1,1]/PQ[2,1]*1.0<BR><BR>The
continued fraction of x^2 is interesting: contfrac(x^2) = </FONT><BR><FONT
face="Times New Roman" size=3>[1,1,4, 1,74, 1,8457, 1,186282390,
1,1,1,2,1,430917181166219, 11,37,1,4,2,</FONT> <BR><FONT
face="Times New Roman" size=3>41151315877490090952542206046,
11,5,3,12,2,34,2,9,8,1,1,2,7,</FONT> <BR><FONT face="Times New Roman"
size=3>13991468824374967392702752173757116934238293984253807017, ...]</FONT>
<BR><FONT face="Times New Roman" size=3>and some of the partial quotients of
x^2 seem to grow exponentially. </FONT><BR><FONT face="Times New Roman"
size=3> </FONT> <BR><FONT face="Times New Roman" size=3>Has anyone seen
this constant before? <BR>I wonder if it has some nice series representations
as well ... <BR></FONT><BR><FONT face="Times New Roman"
size=3>Thanks,<BR> Paul</FONT> </BLOCKQUOTE></BODY></HTML>