login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=1..105.

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-Fran├žois 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: A248511 A210191 A110465 * A094444 A231641 A099446

Adjacent sequences:  A100335 A100336 A100337 * A100339 A100340 A100341

KEYWORD

nonn,cons

AUTHOR

Paul D. Hanna, Nov 17 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 20 03:10 EDT 2019. Contains 323412 sequences. (Running on oeis4.)