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A258075 Constant x that satisfies: x = Sum_{n>=1} frac(n*(1-x)) / 2^n. 1

%I

%S 0,4,4,3,5,7,4,6,9,0,1,5,0,0,3,2,6,1,5,7,8,6,0,4,0,4,4,3,5,7,4,6,9,0,

%T 1,5,0,0,3,2,5,8,5,1,1,4,3,0,2,6,5,3,0,3,8,8,1,2,5,7,7,4,6,2,8,4,3,1,

%U 2,5,3,1,6,4,8,5,9,0,0,2,2,3,4,3,0,2,5,3,7,5,8,8,5,2,7,8,2,4,8,5,9,9,1,9,5,8,2,4,2,9,2,7,1,5,6,9,5,0,4,3,3,1,0,9,7,8,1,3,6,1,1,9,6,0,1,8,3,1,0,4,9,5,5,4,6,2,3,2,2,6,0,2

%N Constant x that satisfies: x = Sum_{n>=1} frac(n*(1-x)) / 2^n.

%C A good approximation to this constant is 680/1533, which is correct to 39 digits.

%H Eric Weisstein, <a href="http://mathworld.wolfram.com/DevilsStaircase.html">Devil's Staircase</a> from MathWorld.

%F This constant satisfies:

%F (1) 1 = x + Sum_{n>=1} {n*x} / 2^n, where {z} denotes the fractional part of z.

%F (2) 1 = 3*x - Sum_{n>=1} [n*x] / 2^n, where [z] denotes the integer floor of z.

%F (3) 1 = 3*x - Sum_{n>=1} 1 / 2^[n/x], a "devil's staircase" sum.

%F (4) 2 = 3*x + Sum_{n>=1} 1 / 2^[n/(1-x)], a "devil's staircase" sum.

%e x = 0.4435746901500326157860404435746901500325851143026...

%e where x = Sum_{n>=1} {n*(1-x)} / 2^n such that x < 1/2 and x > 0.

%e Other series involving x begin:

%e (a) 3*x-1 = 0/2 + 0/2^2 + 1/2^3 + 1/2^4 + 2/2^5 + 2/2^6 + 3/2^7 + 3/2^8 + 3/2^9 + 4/2^10 + 4/2^11 + 5/2^12 + 5/2^13 + 6/2^14 +...+ [n*x]/2^n +...

%e (b) 2-3*x = 0/2 + 1/2^2 + 1/2^3 + 2/2^4 + 2/2^5 + 3/2^6 + 3/2^7 + 4/2^8 + 5/2^9 + 5/2^10 + 6/2^11 + 6/2^12 + 7/2^13 + 7/2^14 +...+ [n*(1-x)]/2^n +...

%e (c) 3*x-1 = 1/2^2 + 1/2^4 + 1/2^6 + 1/2^9 + 1/2^11 + 1/2^13 + 1/2^15 + 1/2^18 + 1/2^20 + 1/2^22 + 1/2^24 + 1/2^27 + 1/2^29 +...+ 1/2^[n/x] +...

%e (d) 2-3*x = 1/2^1 + 1/2^3 + 1/2^5 + 1/2^7 + 1/2^8 + 1/2^10 + 1/2^12 + 1/2^14 + 1/2^16 + 1/2^17 + 1/2^19 + 1/2^21 + 1/2^23 +...+ 1/2^[n/(1-x)] +...

%e note that (c) and (d) involve Beatty sequences as exponents of 1/2.

%e The complement to this constant is A258072:

%e 1-x = 0.5564253098499673842139595564253098499674148856973...

%e and possesses very similar properties.

%e The CONTINUED FRACTION of 3*x has large partial quotients:

%e 3*x = [1, 3, 42, 4, 41619663273108911871743469597819008, 10889035741470030830827987437816582767104, ...];

%e the number of digits of the partial quotients begin:

%e [1, 1, 2, 1, 35, 41, 115, 270, ...].

%e The initial 1050 digits are:

%e x = 0.44357469015003261578604044357469015003258511430265\

%e 30388125774628431253164859002234302537588527824859\

%e 91958242927156950433109781361196018310495546232260\

%e 21308756488922508493146400309145266931418310463207\

%e 21682655581705538047690626270645407928462460161012\

%e 59348074253641658104334540972276437282369203087870\

%e 91688278965124501135824721922196014571344903077739\

%e 18500944407821228291964676297532896809047816970961\

%e 57232314116611612205997186868899794801384103713379\

%e 27093301379877141640033100637461650643685872664216\

%e 71210915983869597104515647764570759665860825582616\

%e 02202386683742093208255873853725828394394746230649\

%e 17646005634235224736778439225789268521850715113549\

%e 81826246824173592162652534369620590399778082914397\

%e 59525545391741573753969070145840793634947962855349\

%e 40161700909555769785319066569343646229986269080724\

%e 03331326445769921119003906381638002817964787631913\

%e 86332217342163925996090891887058572086607050246768\

%e 90125594710389427906358536708700202340858220467421\

%e 83637059019834780162093855868315000246318027150733\

%e 19542694230313602926455823852338591379348725144367...

%o (PARI) {x=.4; for(i=1, 100, x = (x + sum(n=1, 4000, frac(n*(1-x))/2^n*1.))/2); x}

%o (PARI) /* Series 2-3*x = Sum_{n>=1} 1 / 2^[n/(1-x)] gives faster convergence: */

%o {x=0.4; for(i=1, 10, x = (2 - sum(n=1, 2000, 1./2^floor(n/(1-x))))/3 ); x}

%Y Cf. A258072.

%K nonn,cons

%O 1,2

%A _Paul D. Hanna_, May 21 2015

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Last modified September 25 18:51 EDT 2021. Contains 347659 sequences. (Running on oeis4.)