A303561 - OEIS

A303561

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

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

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OFFSET

LINKS

Table of n, a(n) for n=1..121.

FORMULA

This constant may be defined by the following expressions.

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

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

(3) A = Sum_{n>=1} A143862(n)/2^n where A143862(n) = Sum_{d|n} binomial(n/d-1, d-1) for n>=1.

EXAMPLE

In base 2: A = 1.00011101001100100010010001000011111011111011010100...

In base 10: A = 1.1140463510380030496149942362001772475651431655583890...

This constant equals the sum of the following infinite series.

(1) A = 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) A = 1/2 + 5/2^4 + 9^2/2^9 + 17^3/2^16 + 33^4/2^25 + 65^5/2^36 + 129^6/2^49 + 257^7/2^64 + 513^8/2^81 + 1025^9/2^100 + 2049^10/2^121 + 4097^11/2^144 + ...

Expressed in terms of powers of 1/2, we have

(3) A = 1/2 + 1/2^2 + 1/2^3 + 2/2^4 + 1/2^5 + 3/2^6 + 1/2^7 + 4/2^8 + 2/2^9 + 5/2^10 + 1/2^11 + 9/2^12 + 1/2^13 + 7/2^14 + 7/2^15 + 9/2^16 + 1/2^17 + 19/2^18 + 1/2^19 + 14/2^20 + 16/2^21 + 11/2^22 + ... + A143862(n)/2^n + ...

BINARY EXPANSION TO 2000 DIGITS:

A = 1.00011101001100100010010001000011111011111011010100\

11010100000101111010110101001100110001110101010111\

10000011001011110111000100111100101111000111011010\

01010111111100100111111110101100101001101000001100\

10101010011111011100101001011100001000110010101010\

10110000011110000010101010011010110101101101100010\

11111110101100010110110010000101110001011010010110\

10100100000010001101011111011110000010001011010101\

11001110000111101001001101011011100101111110101101\

01101001010011101011010001110010001110100011000100\

10101010001001100100000111110000000110010101000000\

00101011010111011111010000000101111010000000010000\

11110110100001000010110000101110101000101110000100\

11001001001010111010111101111110000110111100100001\

10011111011011111100010101110110001001101000011001\

10111010010111111011110110001000101101110110100100\

11100100001101000000000100010010111100000110110001\

00010100011000000000110001110100111101011001110010\

10101100011110000110100110101000011110101000011101\

01111111100010111010111101011001001010000011100110\

11011101010111010011011001000111101111000110110101\

01000110011010101101101111001001000001111001110101\

11110010011100011001011011000110001100000001010010\

00010000100101010001100001000001011101010001100000\

01110010010111010011011101000011101000100000100100\

10111000110001100101000100001100010111001010011111\

11111000101111010110111010101110110111111001001000\

11110110001100011010110010100110101011000110101110\

01000011111010111010100101001111001011011111001001\

10011111110000011010010011010100011001010100100111\

11100000100000100011110000010101011110100010000000\

10101100000010010001000100001101111111101100100111\

11110100111010110111001010111010101111110111110111\

00101000101001001000001110111011011000011101111101\

11000100101010001010011110110111100110100100100010\

10010111001111101001000100100000001001110100011111\

11100101101110100001101101111111101101100100110100\

00000100000010011111100100111000110101110101110000\

11001111111100010010000000011101100011110100111110\

11100001011010100011001010010011010110001101111100...

CROSSREFS

Cf. A303560, A143862.

Sequence in context: A120532 A373156 A320106 * A284680 A004555 A372573

Adjacent sequences: A303558 A303559 A303560 * A303562 A303563 A303564

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 26 2018

STATUS

approved