This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A317413 Continued fraction for binary expansion of Liouville's number interpreted in base 2 (A012245). 6

%I

%S 0,1,3,3,1,2,1,4095,3,1,3,3,1,4722366482869645213695,4,3,1,3,4095,1,2,

%T 1,3,3,1

%N Continued fraction for binary expansion of Liouville's number interpreted in base 2 (A012245).

%C The continued fraction of the number obtained by reading A012245 as binary fraction.

%C Except for the first term, the only values that occur in this sequence are 1, 2, 3, 4 and values 2^((m-1)*m!) - 1 for m > 2. The probability of occurence P(a(n) = k) are given by:

%C P(a(n) = 1) = 1/3,

%C P(a(n) = 2) = 1/12,

%C P(a(n) = 3) = 1/3,

%C P(a(n) = 4) = 1/12 and

%C P(a(n) = 2^((m-1)*m!)-1) = 1/(3*2^(m-1)) for m > 2.

%C The next term is roughly 3.12174855*10^144 (see b-file for precise value).

%H A.H.M. Smeets, <a href="/A317413/b317413.txt">Table of n, a(n) for n = 0..48</a>

%F a(n) = 1 if and only if n in A317538.

%F a(n) = 2 if and only if n in {24*m - 19 | m > 0} union {24*m - 4 | m > 0}.

%F a(n) = 3 if and only if n in A317539.

%F a(n) = 4 if and only if n in {12*m + A014710(m-1) - 2*(A014710(m-1) mod 2) | m > 0}

%F a(n) = 2^((m-1)*m!)-1 if and only if n in {3*2^(m-2)*(1+k*4) - 1 | k >= 0} union {3*2^m-2)*(3+k*4) | k >= 0} for m > 2.

%p with(numtheory): cfrac(add(1/2^factorial(n),n=1..7),24,'quotients'); # _Muniru A Asiru_, Aug 11 2018

%t ContinuedFraction[ FromDigits[ RealDigits[ Sum[1/10^n!, {n, 8}], 10, 10000], 2], 60] (* _Robert G. Wilson v_, Aug 09 2018 *)

%o (Python)

%o n,f,i,p,q,base = 1,1,0,0,1,2

%o while i < 100000:

%o ....i,p,q = i+1,p*base,q*base

%o ....if i == f:

%o ........p,n = p+1,n+1

%o ........f = f*n

%o n,a,j = 0,0,0

%o while p%q > 0:

%o ....a,f,p,q = a+1,p//q,q,p%q

%o ....print(a-1,f)

%Y Cf. A012245, A014710, A317538, A317539.

%Y Cf. A058304 (in base 10), A317414 (in base 3).

%K nonn,base,cofr

%O 0,3

%A _A.H.M. Smeets_, Jul 27 2018

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.

Last modified December 11 18:34 EST 2019. Contains 329925 sequences. (Running on oeis4.)