A100338 Decimal expansion of the constant x whose continued fraction expansion equals A006519 (highest power of 2 dividing n).
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
Examples
1.353871128429882374388894084016608124227333416812118556923672649787001842...
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.
Programs
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Mathematica
cf = ContinuedFraction[ Table[ 2^IntegerExponent[n, 2], {n, 1, 200}]]; RealDigits[ FromContinuedFraction[cf // Flatten] , 10, 105] // First (* Jean-François Alcover, Feb 19 2013 *) -
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}
Comments