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

Author

Paul D. Hanna, Nov 17 2004

Keywords

• nonn
• cons

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.

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.

Crossrefs

Cf. A006519, A100863, A100864, A100865.

Programs

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}