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A265222 Decimal expansion of the least real z > 1 that satisfies:  1/2 = Sum_{n>=1} {z^n} / 2^n, where {x} denotes the fractional part of x. 0
1, 3, 5, 5, 4, 9, 3, 0, 5, 1, 8, 4, 3, 4, 6, 3, 9, 6, 0, 4, 3, 0, 7, 9, 8, 2, 5, 4, 5, 1, 3, 6, 0, 1, 7, 2, 1, 5, 3, 6, 7, 9, 8, 6, 2, 6, 6, 2, 3, 0, 4, 0, 1, 7, 0, 5, 9, 7, 0, 3, 9, 9, 8, 2, 5, 1, 8, 3, 4, 6, 4, 5, 9, 5, 9, 6, 5, 6, 7, 9, 9, 1, 1, 7, 5, 2, 5, 8, 4, 1, 5, 8, 5, 9, 2, 4, 2, 4, 1, 6, 0, 2, 7, 8, 1, 0, 7, 7, 7, 2, 2, 0, 8, 1, 0, 3, 7, 6, 2, 7, 5, 5, 4, 0, 9, 0, 4, 7, 2, 4, 4, 6, 2, 6, 1, 8, 3, 7, 5, 9, 2, 7, 7, 3, 1, 9, 1, 7, 0, 5, 6, 5, 4, 5, 5, 0, 4, 3, 0, 5, 1, 1, 5, 5, 1, 5, 2, 5, 3, 2, 5, 3, 5, 7, 6, 4, 6, 8, 8, 4, 8, 0, 5, 5, 2, 7, 8, 2, 5, 8, 8, 2, 5, 4, 8, 5, 2, 4, 6, 8, 6, 8, 0, 7, 3, 6, 7, 0, 0, 2, 5, 8, 9, 1, 3, 7, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Compare to the trivial sum: 1/2 = Sum_{n>=1} {z^n} / 2^n when z = 2/3.

Are there an infinite number of solutions to z such that 1/2 = Sum_{n>=1} {z^n}/2^n for z in the interval (1,2)?  Can it be shown that these solutions are irrational?

LINKS

Table of n, a(n) for n=1..217.

FORMULA

Constant z satisfies:  z/(2-z) = 1/2 + Sum_{n>=1} floor(z^n) / 2^n.

EXAMPLE

z = 1.35549305184346396043079825451360172153679862662304\

01705970399825183464595965679911752584158592424160\

27810777220810376275540904724462618375927731917056\

54550430511551525325357646884805527825882548524686\

80736700258913704221335857525909384098830553851116\

05126668310187108162998498933972248896649516339678\

49366846073141291529984290772350906229012817370186\

55169666380457011989017081864719838727980967224971\

98474075748848718792660615793372688507739995250655\

89552283138883099708481229931272775563504550828971\

70024113427009808993763831069451402518903559858745\

59505239672622181099687153202444348446965100196443\

80668334503687174824625643625205785168626890858603\

93558591010025659573835216359698255844450783545599\

17427663656403996437836675351715525954403386934996\

84666455482949770524207287282642519434321755773398\

62694330881627744201473062989839201202400657816150\

84391198373759362968161495215799374878373246239017\

13574353581079553116458694020174184103284836868862\

62896573800985278249325536552468829105057478796554...

GENERATING METHOD.

Set z = 1, then 5*N iterations of: z = z + 1/6 - 1/3*suminf(n=1, frac(z^n)/2^n ) yields about N digits.

RELATED CONSTANTS.

(1) Least real z > sqrt(2) such that 1/2 = Sum_{n>=1} {z^n}/2^n is

z = 1.53249422831624633589209045977055204808626394342423\

14743943812327455836621055716194661407233767648443884690...

(2) Least real z > 13^(1/6) such that 1/2 = Sum_{n>=1} {z^n}/2^n is

z = 1.53687788811637150697634883884345836398432123973966\

82723170954185424082387135095296328278152477537914182407...

(3) Least real z > 113^(1/11) such that 1/2 = Sum_{n>=1} {z^n}/2^n is

z = 1.53695074329851152802228619086712434244494261099208\

23135334501884406111892113321815249303292032820021290733...

(4) Least real z > 4^(1/3) such that 1/2 = Sum_{n>=1} {z^n}/2^n is

z = 1.59632077942330242620034231916724745224383614542308\

85710643389591659503459032792156507140054529416666027413...

(5) Least real z > sqrt(3) such that 1/2 = Sum_{n>=1} {z^n}/2^n is

z = 1.75118759092320930011149976687513099251284547995833\

95627867298408820204655832368182553275152837639807086641...

(6) Least real z > 73562^(1/20) such that 1/2 = Sum_{n>=1} {z^n}/2^n is

z = 1.75118762663466598663070625956863995162241902713017\

79945479735132207866837514950592202363248785148839017212...

(7) Least real z > 691806^(1/24) such that 1/2 = Sum_{n>=1} {z^n}/2^n is

z = 1.75118762870531088915040587334001786652479560768473\

66794998600900660167466413594621357623811851518779554331...

(8) Least real z > 13698^(1/17) such that 1/2 = Sum_{n>=1} {z^n}/2^n is

z = 1.75118802293006688271680253510886080565943343511207\

61212797559379720869345357325961787461767015435821839729...

(9) Least real z > 475^(1/11) such that 1/2 = Sum_{n>=1} {z^n}/2^n is

z = 1.75121702526493397702106373116694543383377610186430\

88066729332431554693339104712708810382658857189983724634...

(10) Least real z > 832^(1/12) such that 1/2 = Sum_{n>=1} {z^n}/2^n is

z = 1.75123699808590302107034621374375230412694312101003\

33815109181242003520899218074396156255274527703355463573...

It not known where the threshold values lie for all z in the interval (1,2).

PROG

(PARI) N=100 \\ Calculate and Print N digits of the constant

\p500 \\ set precision

{z=1.0; for(i=1, 5*N, z = z + 1/6 - 1/3*suminf(n=1, frac(z^n)/2^n ) ); z}

{a(n) = floor((10^n*z))%10}

{m=0; for(n=0, N, print1( floor((10^n*z))%10, ", "); if(m==50, m=0; print("")); m=m+1)}

CROSSREFS

Sequence in context: A153098 A119280 A307634 * A160585 A307447 A016658

Adjacent sequences:  A265219 A265220 A265221 * A265223 A265224 A265225

KEYWORD

nonn,cons

AUTHOR

Paul D. Hanna, Dec 05 2015

STATUS

approved

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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)