# [seqfan] Constant with 2-Adic Valuation Continued Fraction

Paul D. Hanna pauldhanna at juno.com
Wed Nov 17 22:48:51 CET 2004

```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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