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A265274 Least real z > 1/2 such that 1/2 = Sum_{n>=1} {n*z} / 2^n, where {x} denotes the fractional part of x. 5
5, 8, 8, 7, 0, 9, 5, 5, 4, 3, 6, 3, 6, 6, 5, 4, 9, 4, 2, 7, 4, 0, 9, 5, 7, 1, 9, 1, 1, 4, 0, 6, 7, 9, 4, 7, 9, 0, 6, 0, 9, 6, 8, 7, 5, 0, 5, 1, 5, 9, 0, 4, 8, 4, 8, 9, 5, 5, 9, 2, 1, 5, 5, 2, 0, 3, 9, 0, 2, 8, 0, 4, 1, 6, 6, 5, 4, 7, 5, 7, 7, 1, 0, 5, 0, 8, 5, 8, 7, 3, 2, 5, 8, 3, 0, 5, 3, 6, 2, 9, 2, 9, 1, 5, 4, 1, 5, 3, 1, 4, 4, 4, 5, 0, 9, 7, 8, 8, 5, 3, 9, 6, 8, 8, 6, 0, 4, 4, 4, 4, 8, 0, 5, 7, 4, 4, 6, 4, 8, 7, 8, 3, 2, 6, 1, 3, 6, 1, 3, 4, 4, 8, 1, 6, 6, 4, 0, 0, 1, 3, 9, 2, 7, 7, 9, 7, 2, 3, 0, 6, 3, 8, 4, 6, 2, 1, 0, 0, 0, 7, 5, 0, 2, 0, 8, 2, 3, 1, 0, 7, 9, 2, 0, 6, 2, 6, 7, 0, 5, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This constant is transcendental.

The rational approximation z ~ 50081870146959747811507711449530545577/85070591730234615865843651857942052860 is accurate to many thousands of 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 A265273.

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 = 0.58870955436366549427409571911406794790609687505159048489559215520...

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; 1, 1, 2, 3, 7, 528, 2, 1, 1, 1, 20282564347337181724466999721987, 2, 1, 2, ...]

(the next partial quotient has too many digits to show).

The convergents of the continued fraction of z begin:

[0/1, 1/1, 1/2, 3/5, 10/17, 73/124, 38554/65489, 77181/131102, 115735/196591, 192916/327693, 308651/524284, 6260233768369968476438463931191202453/10633823966279326983230456482242560001, ...]

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

[0; 1, 2, 10, 4228, 162260514778697453795735997775904, ..., Q_n, ...]

where

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

Q_2 : 2^1*(2^(1*1) - 1)/(2^1 - 1) = 2;

Q_3 : 2^1*(2^(2*2) - 1)/(2^2 - 1) = 10;

Q_4 : 2^2*(2^(3*5) - 1)/(2^5 - 1) = 4228;

Q_5 : 2^5*(2^(7*17) - 1)/(2^17 - 1) = 162260514778697453795735997775904;

Q_6 : 2^17*(2^(528*124) - 1)/(2^124 - 1) ;

Q_7 : 2^124*(2^(2*65489) - 1)/(2^65489 - 1) ;

Q_8 : 2^65489*(2^(1*131102) - 1)/(2^131102 - 1) ;

Q_9 : 2^131102*(2^(1*196591) - 1)/(2^196591 - 1) ;

Q_10 : 2^196591*(2^(1*327693) - 1)/(2^327693 - 1) ;

Q_10 = 2^327693*(2^(20282564347337181724466999721987*524284) - 1)/(2^524284 - 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. A265271, A265272, A265273, A265275, A265276.

Sequence in context: A069997 A199373 A247037 * A200286 A073822 A198606

Adjacent sequences:  A265271 A265272 A265273 * A265275 A265276 A265277

KEYWORD

nonn,cons

AUTHOR

Paul D. Hanna, Dec 12 2015

STATUS

approved

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Last modified November 14 20:15 EST 2019. Contains 329130 sequences. (Running on oeis4.)