

A265273


Least positive real z > 2/5 such that 1/2 = Sum_{n>=1} {n*z} / 2^n, where {x} denotes the fractional part of x.


5



4, 1, 1, 2, 9, 0, 4, 4, 5, 6, 3, 6, 3, 3, 4, 5, 0, 5, 7, 2, 5, 9, 0, 4, 2, 8, 0, 8, 8, 5, 9, 3, 2, 0, 5, 2, 0, 9, 3, 9, 0, 3, 1, 2, 4, 9, 4, 8, 4, 0, 9, 5, 1, 5, 1, 0, 4, 4, 0, 7, 8, 4, 4, 7, 9, 6, 0, 9, 7, 1, 9, 5, 8, 3, 3, 4, 5, 2, 4, 2, 2, 8, 9, 4, 9, 1, 4, 1, 2, 6, 7, 4, 1, 6, 9, 4, 6, 3, 7, 0, 7, 0, 8, 4, 5, 8, 4, 6, 8, 5, 5, 5, 4, 9, 0, 2, 1, 1, 4, 6, 0, 3, 1, 1, 3, 9, 5, 5, 5, 5, 1, 9, 4, 2, 5, 5, 3, 5, 1, 2, 1, 6, 7, 3, 8, 6, 3, 8, 6, 5, 5, 1, 8, 3, 3, 5, 9, 9, 8, 6, 0, 7, 2, 2, 0, 2, 7, 6, 9, 3, 6, 1, 5, 3, 7, 8, 9, 9, 9, 2, 4, 9, 7, 9, 1, 7, 6, 8, 9, 2, 0, 7, 9, 3, 7, 3, 2, 9, 4, 3
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

This constant is transcendental.
The rational approximation z ~ 34988721583274868054335940408411507283/85070591730234615865843651857942052860 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 crossreferences for other solutions.
The complement to this constant is given by A265274.


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/(1z)],
(4) 3/2  2*z = Sum_{n>=1} [n*(1z)] / 2^n,
(5) 1/2 = Sum_{n>=1} {n*(1z)} / 2^n,
where [x] denotes the integer floor function of x.


EXAMPLE

z = 0.411290445636334505725904280885932052093903124948409515104407844796...
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; 2, 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, 1/2, 2/5, 7/17, 51/124, 26935/65489, 53921/131102, 80856/196591, 134777/327693, 215633/524284, 4373590197909358506791992551051357548/10633823966279326983230456482242560001, ...]
The partial quotients of the continued fraction of 2*z  1/2 are as follows:
[0; 3, 10, 4228, 162260514778697453795735997775904, ..., Q_n, ...]
where
Q_1 = 2^0*(2^(2*1)  1)/(2^1  1) = 3 ;
Q_2 = 2^1*(2^(2*2)  1)/(2^2  1) = 10 ;
Q_3 = 2^2*(2^(3*5)  1)/(2^5  1) = 4228 ;
Q_4 = 2^5*(2^(7*17)  1)/(2^17  1) = 162260514778697453795735997775904 ;
Q_5 = 2^17*(2^(528*124)  1)/(2^124  1) ;
Q_6 = 2^124*(2^(2*65489)  1)/(2^65489  1) ;
Q_7 = 2^65489*(2^(1*131102)  1)/(2^131102  1) ;
Q_8 = 2^131102*(2^(1*196591)  1)/(2^196591  1) ;
Q_9 = 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, A265274, A265275, A265276.
Sequence in context: A279206 A016527 A010325 * A293770 A111311 A327893
Adjacent sequences: A265270 A265271 A265272 * A265274 A265275 A265276


KEYWORD

nonn,cons


AUTHOR

Paul D. Hanna, Dec 09 2015


STATUS

approved



