A303340 Binary expansion of constant: B = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.

1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0

Offset

0

Links

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

Formula

This constant may be defined by the following expressions.

(1) B = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.

(2) B = Sum_{n>=1} (2^n - 1)^(n-1) / 2^(n^2).

(3) B = Sum_{n>=1} A303506(n)/2^n where A303506(n) = Sum_{d|n} binomial(n/d-1, d-1) * (-1)^(d-1) for n>=1.

Example

In base  2: B = 0.11100100010010111111111100111101001100001001001100...
In base 10: B = 0.891784622610953349715890136060239421022216970366139...
This constant equals the sum of the following infinite series.
(1) B = 1 - 1/3^2 + 1/7^3 - 1/15^4 + 1/31^5 - 1/63^6 + 1/127^7 - 1/255^8 + 1/511^9 - 1/1023^10 + 1/2047^11 - 1/4095^12 + 1/8191^13 - 1/16383^14 + ...
Also,
(2) B = 1/2 + 3/2^4 + 7^2/2^9 + 15^3/2^16 + 31^4/2^25 + 63^5/2^36 + 127^6/2^49 + 255^7/2^64 + 511^8/2^81 + 1023^9/2^100 + 2047^10/2^121 + 4095^11/2^144 + ...
Expressed in terms of powers of 1/2, we have
(3) B = 1/2 + 1/2^2 + 1/2^3 + 0/2^4 + 1/2^5 - 1/2^6 + 1/2^7 - 2/2^8 + 2/2^9 - 3/2^10 + 1/2^11 - 1/2^12 + 1/2^13 - 5/2^14 + 7/2^15 - 7/2^16 + 1/2^17 + 3/2^18 + 1/2^19 - 12/2^20 + 16/2^21 - 9/2^22 + ... + A303506(n)/2^n + ...
The binary expansion of this constant begins:
B = 0.11100100010010111111111100111101001100001001001100\
11111010000100110110010100000011000111010001001111\
01010101000100000011111100001011011110100001111110\
00011100101101000101010111011000011001000010101000\
01001101100101010110001001110110001101010000111011\
11110111011111000101010011100001111110011111011011\
00001001000101000101100000000010101111100010001111\
10100001111001110011001100011110110001010001100100\
10101000101110001000111001010110110111101010011011\
00100111110000000001101111110010000010110001111101\
00101101000000110001000010100011111001100010100001\
10100000001011000111111010111100100110100101011101\
01010000001001000100000011101111110011001010001011\
00000111011010111010100000110110111111111100111001\
11011011110001111100010010011111000010000100001011\
00011101110101100100011001101110011001111111001001\
10011101110010111110111111010110101110001110001000\
00111110001010100101111000110010101011000101000111\
10111001011111000000101000111001001010100100010000\
00111111111001110011101011111110010000000010111100\
11111110000100100011101010110100100010001100010111\
10000110111011001111011000100001101110100100001000\
00010000010111011111011000000011100110100100111111\
00110000010111110010110011001111010110011001000101\
11110001000111111001111001100001001100000101010000\
01100100111000100011010000101000100101100100111010\
00100111111011110010101110000101010110110101000010\
10111001001110101101100010101110010110011010100100\
00001110111111000000101111000111000110110111111111\
10000010011101001001010101000001001010000010101111\
11000100010111110001011011101100011001011000101101\
00010010001010000011110101111000110100001101010111\
00000111111001000011000100111101110101001111110110\
01000111010110100010100010101000110000100101001001\
11111101110111001100011111111100100000011110001100\
01111011011100011001010100110010110010011100111001\
11011100010110001100010001101110111101100110111111\
10011110000010100100100010010011011110110010111100\
00110101110110110110100001110011110111110001101110\
11010011010111010110001000000100101110011010110010...

Mathematica

bits = 200; B = NSum[(-1)^(n-1)/(2^n-1)^n, {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> Ceiling[Log[10, 2]*bits]]; RealDigits[B, 2, bits][[1]] (* Jean-François Alcover, Apr 25 2018 *)

Crossrefs

Cf. A302765 (decimal expansion), A303506.

Sequence in context: A351114 A128190 A128189 * A115512 A115513 A133080

Adjacent sequences: A303337 A303338 A303339 * A303341 A303342 A303343

Keyword

nonn,cons

Author

Paul D. Hanna, Apr 24 2018

Status

approved