A258072 Constant x that satisfies: x = Sum_{n>=1} frac(n*(1-x)) / 2^n.

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

Offset

1,2

Comments

A good approximation to this constant is 853/1533, which is correct to 39 digits.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1051

Eric Weisstein, Devil's Staircase from MathWorld.

Formula

This constant satisfies:

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

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

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

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

Example

x = 0.5564253098499673842139595564253098499674148856973...
where x = Sum_{n>=1} {n*(1-x)} / 2^n such that x > 1/2 and x < 1.
Other series involving x begin:
(a) 3*x-1 = 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*x]/2^n +...
(b) 2-3*x = 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*(1-x)]/2^n +...
(c) 3*x-1 = 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/x] +...
(d) 2-3*x = 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/(1-x)] +...
note that (c) and (d) involve Beatty sequences as exponents of 1/2.
The complement to this constant is A258075:
1-x = 0.4435746901500326157860404435746901500325851143026...
and possesses very similar properties.
The CONTINUED FRACTION of this constant has large partial quotients:
x = [0, 1, 1, 3, 1, 13, 2, 2, 2, 13873221091036303957247823199273002, 2, 1, 3629678580490010276942662479272194255700, 1, 2, 1641750258183103300511626670839317241878322469602726944497845558510238407145724198024905280171526972004597661433855, 1, 2];
the next partial quotient has 269 digits.
The initial 1050 digits are:
x = 0.5564253098499673842139595564253098499674148856973\
46961187422537156874683514099776569746241147217514\
00804175707284304956689021863880398168950445376773\
97869124351107749150685359969085473306858168953679\
27831734441829446195230937372935459207153753983898\
74065192574635834189566545902772356271763079691212\
90831172103487549886417527807780398542865509692226\
08149905559217877170803532370246710319095218302903\
84276768588338838779400281313110020519861589628662\
07290669862012285835996689936253834935631412733578\
32878908401613040289548435223542924033413917441738\
39779761331625790679174412614627417160560525376935\
08235399436576477526322156077421073147814928488645\
01817375317582640783734746563037940960022191708560\
24047445460825842624603092985415920636505203714465\
05983829909044423021468093343065635377001373091927\
59666867355423007888099609361836199718203521236808\
61366778265783607400390910811294142791339294975323\
10987440528961057209364146329129979765914177953257\
81636294098016521983790614413168499975368197284926\
68045730576968639707354417614766140862065127485563...

Prog

(PARI) {x=.5; for(i=1, 100, x = (x + sum(n=1, 4000, frac(n*(1-x))/2^n*1.))/2); x}
(PARI) /* Series 2-3*x = Sum_{n>=1} 1 / 2^[n/(1-x)] gives faster convergence: */
{x=.5; for(i=1, 10, x = (2 - sum(n=1, 2000, 1./2^floor(n/(1-x))))/3 ); x}

Crossrefs

Cf. A258075.

Sequence in context: A193555 A161981 A346962 * A091284 A276860 A046604

Adjacent sequences: A258069 A258070 A258071 * A258073 A258074 A258075

Keyword

nonn,cons

Author

Paul D. Hanna, May 18 2015

Status

approved