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A100338
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Decimal expansion of the constant x whose continued fraction expansion equals A006519 (highest power of 2 dividing n).
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14
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1, 3, 5, 3, 8, 7, 1, 1, 2, 8, 4, 2, 9, 8, 8, 2, 3, 7, 4, 3, 8, 8, 8, 9, 4, 0, 8, 4, 0, 1, 6, 6, 0, 8, 1, 2, 4, 2, 2, 7, 3, 3, 3, 4, 1, 6, 8, 1, 2, 1, 1, 8, 5, 5, 6, 9, 2, 3, 6, 7, 2, 6, 4, 9, 7, 8, 7, 0, 0, 1, 8, 4, 2, 0, 6, 4, 8, 2, 6, 0, 5, 4, 8, 4, 3, 1, 9, 6, 9, 7, 6, 0, 1, 7, 4, 6, 5, 6, 9, 7, 9, 6, 6, 8, 5
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OFFSET
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1,2
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COMMENTS
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This constant x has the special property that the continued fraction expansion of 2*x results in the continued fraction expansion of x interleaved with 2's: contfrac(x) = [1;2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,...A006519(n),... ] while contfrac(2*x) = [2;1, 2,2, 2,1, 2,4, 2,1, 2,2, 2,1, 2,8,... 2, A006519(n),...].
The continued fraction of x^2 has large partial quotients (see A100864, A100865) that appear to be doubly exponential.
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LINKS
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Table of n, a(n) for n=1..105.
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EXAMPLE
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x=1.353871128429882374388894084016608124227333416812118556923672649787001842...
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MATHEMATICA
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cf = ContinuedFraction[ Table[ 2^IntegerExponent[n, 2], {n, 1, 200}]]; RealDigits[ FromContinuedFraction[cf // Flatten] , 10, 105] // First (* Jean-François Alcover, Feb 19 2013 *)
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PROG
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(PARI) /* This PARI code generates 1000 digits of x very quickly: */ {x=sqrt(2); y=x; L=2^10; for(i=1, 10, v=contfrac(x, 2*L); if(2*L>#v, v=concat(v, vector(2*L-#v+1, j, 1))); if(2*L>#w, w=concat(w, vector(2*L-#w+1, j, 1))); w=vector(2*L, n, if(n%2==1, 2, w[n]=v[n\2])); w[1]=floor(2*x); CFW=contfracpnqn(w); x=CFW[1, 1]/CFW[2, 1]*1.0/2; ); x}
(PARI) {CFM=contfracpnqn(vector(1500, n, 2^valuation(n, 2))); x=CFM[1, 1]/CFM[2, 1]*1.0}
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CROSSREFS
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Cf. A006519, A100863, A100864, A100865.
Sequence in context: A210041 A210191 A110465 * A094444 A099446 A198827
Adjacent sequences: A100335 A100336 A100337 * A100339 A100340 A100341
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KEYWORD
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nonn,cons
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AUTHOR
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Paul D. Hanna, Nov 17 2004
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STATUS
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approved
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