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A265271 Least positive real z such that 1/2 = Sum_{n>=1} {n*z} / 2^n, where {x} denotes the fractional part of x. 5
3, 2, 1, 4, 2, 8, 5, 6, 9, 2, 9, 9, 8, 3, 4, 1, 0, 7, 2, 3, 4, 4, 5, 6, 0, 4, 4, 5, 6, 3, 8, 5, 9, 8, 6, 7, 1, 6, 9, 9, 3, 1, 0, 3, 6, 1, 2, 1, 8, 8, 6, 3, 5, 8, 1, 1, 9, 1, 2, 4, 0, 1, 8, 0, 9, 9, 6, 2, 1, 0, 0, 5, 7, 2, 7, 4, 2, 8, 9, 6, 4, 2, 5, 5, 1, 1, 3, 0, 2, 1, 4, 8, 9, 6, 5, 3, 8, 1, 6, 4, 0, 8, 1, 1, 9, 4, 1, 1, 7, 9, 6, 7, 7, 6, 2, 4, 9, 2, 4, 7, 7, 0, 0, 9, 0, 4, 4, 8, 7, 4, 4, 9, 3, 1, 9, 9, 8, 6, 4, 3, 7, 7, 0, 8, 0, 8, 8, 8, 9, 6, 0, 8, 1, 1, 8, 2, 7, 1, 8, 5, 7, 9, 4, 0, 6, 7, 3, 2, 9, 8, 9, 1, 2, 7, 6, 8, 4, 3, 4, 4, 0, 8, 1, 8, 9, 4, 8, 4, 5, 0, 7, 5, 5, 1, 3, 5, 9, 0, 4, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This constant is transcendental.

The rational approximation z ~ 345131297/1073741820 is accurate to over 5 million digits.

This constant is one of 6 solutions to the equation 1/2 = Sum_{n>=1} {n*z}/2^n, where z is in the interval (0,1) - see cross-references for other solutions.

The complement to this constant is given by A265276.

LINKS

Table of n, a(n) for n=0..199.

Eric Weisstein's World of Mathematics, Devil's Staircase.

Index entries for transcendental numbers

FORMULA

The constant z satisfies:

(1) 2*z - 1/2 = Sum_{n>=1} [n*z] / 2^n,

(2) 2*z - 1/2 = Sum_{n>=1} 1 / 2^[n/z],

(3) 3/2 - 2*z = Sum_{n>=1} 1 / 2^[n/(1-z)],

(4) 3/2 - 2*z = Sum_{n>=1} [n*(1-z)] / 2^n,

(5) 1/2 = Sum_{n>=1} {n*(1-z)} / 2^n,

where [x] denotes the integer floor function of x.

EXAMPLE

z = 0.321428569299834107234456044563859867169931036121886358119124...

where z satisfies

(0) 1/2 = {z}/2 + {2*z}/2^2 + {3*z}/2^3 + {4*z}/2^4 + {5*z}/2^5 +...

(1) 2*z - 1/2 = [z]/2 + [2*z]/2^2 + [3*z]/2^3 + [4*z]/2^4 + [5*z]/2^5 +...

(2) 2*z - 1/2 = 1/2^[1/z] + 1/2^[2/z] + 1/2^[3/z] + 1/2^[4/z] + 1/2^[5/z] +...

The continued fraction of the constant z begins:

[0; 3, 8, 1, 599185, 2, 1, 1, 3, 1, 2, ...]

(the next partial quotient has over 5 million digits).

The convergents of the continued fraction of z begin:

[0/1, 1/3, 8/25, 9/28, 5392673/16777205, 10785355/33554438, 16178028/50331643, 26963383/83886081, 97068177/301989886, 124031560/385875967, 345131297/1073741820, ...].

The partial quotients of the continued fraction of 2*z - 1/2 are as follows:

[0; 7, 4793490, 8, ..., Q_n, ...]

where

Q_1 = 2^0*(2^(3*1) - 1)/(2^1 - 1) = 7 ;

Q_2 = 2^1*(2^(8*3) - 1)/(2^3 - 1) = 4793490 ;

Q_3 = 2^3*(2^(1*25) - 1)/(2^25 - 1) = 8 ;

Q_4 = 2^25*(2^(599185*28) - 1)/(2^28 - 1) ;

Q_5 = 2^28*(2^(2*16777205) - 1)/(2^16777205 - 1) = 2^28*(2^16777205 + 1) ;

Q_6 = 2^16777205*(2^(1*33554438) - 1)/(2^33554438 - 1) = 2^16777205 ;

Q_7 = 2^33554438*(2^(1*50331643) - 1)/(2^50331643 - 1) = 2^33554438 ;

Q_8 = 2^50331643*(2^(3*83886081) - 1)/(2^83886081 - 1) ;

Q_9 = 2^83886081*(2^(1*301989886) - 1)/(2^301989886 - 1) ;

Q_10 = 2^301989886*(2^(2*385875967) - 1)/(2^385875967 - 1) ; ...

These partial quotients can be calculated from the simple continued fraction of z and the denominators in the convergents of the continued fraction of z; see the Mathworld link entitled "Devil's Staircase" for more details.

CROSSREFS

Cf. A265272, A265273, A265274, A265275, A265276.

Sequence in context: A112603 A097294 A060848 * A006020 A294177 A224381

Adjacent sequences:  A265268 A265269 A265270 * A265272 A265273 A265274

KEYWORD

nonn,cons

AUTHOR

Paul D. Hanna, Dec 08 2015

STATUS

approved

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Last modified November 14 04:56 EST 2019. Contains 329110 sequences. (Running on oeis4.)