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A100338
Decimal expansion of the constant x whose continued fraction expansion equals A006519 (highest power of 2 dividing n).
14
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
OFFSET
1,2
COMMENTS
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.
LINKS
Dzmitry Badziahin and Jeffrey Shallit, An Unusual Continued Fraction, arXiv:1505.00667 [math.NT], 2015.
Dzmitry Badziahin and Jeffrey Shallit, An unusual continued fraction, Proc. Amer. Math. Soc. 144 (2016), 1887-1896.
EXAMPLE
1.353871128429882374388894084016608124227333416812118556923672649787001842...
MATHEMATICA
cf = ContinuedFraction[ Table[ 2^IntegerExponent[n, 2], {n, 1, 200}]]; RealDigits[ FromContinuedFraction[cf // Flatten] , 10, 105] // First (* Jean-François Alcover, Feb 19 2013 *)
PROG
(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}
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Paul D. Hanna, Nov 17 2004
STATUS
approved