A265271 - OEIS

A265271

Least positive real z such that 1/2 = Sum_{n>=1} {n*z} / 2^n, where {x} denotes the fractional part of x.

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

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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 * A350499 A006020 A294177

Adjacent sequences: A265268 A265269 A265270 * A265272 A265273 A265274

KEYWORD

nonn,cons

AUTHOR

Paul D. Hanna, Dec 08 2015

STATUS

approved