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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 (list; constant; graph; refs; listen; history; text; internal format)
 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, Jeffrey Shallit, An Unusual Continued Fraction, arXiv:1505.00667 [math.NT], 2015. EXAMPLE x=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 Cf. A006519, A100863, A100864, A100865. Sequence in context: A210191 A110465 A328915 * A094444 A231641 A099446 Adjacent sequences:  A100335 A100336 A100337 * A100339 A100340 A100341 KEYWORD nonn,cons AUTHOR Paul D. Hanna, Nov 17 2004 STATUS approved

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Last modified October 1 08:39 EDT 2020. Contains 337442 sequences. (Running on oeis4.)