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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 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.)