A302765 Decimal expansion of constant: B = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.

8, 9, 1, 7, 8, 4, 6, 2, 2, 6, 1, 0, 9, 5, 3, 3, 4, 9, 7, 1, 5, 8, 9, 0, 1, 3, 6, 0, 6, 0, 2, 3, 9, 4, 2, 1, 0, 2, 2, 2, 1, 6, 9, 7, 0, 3, 6, 6, 1, 3, 9, 1, 8, 9, 3, 3, 6, 8, 2, 2, 3, 6, 0, 1, 2, 7, 6, 1, 2, 2, 3, 7, 8, 1, 7, 5, 4, 4, 4, 5, 5, 8, 3, 9, 6, 7, 8, 6, 4, 6, 3, 8, 6, 1, 7, 6, 3, 7, 1, 0, 5, 7, 4, 3, 9, 0, 9, 3, 8, 3, 6, 1, 3, 9, 3, 4, 3, 9, 5, 9

Offset

0,1

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1030

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

Constant B = 0.891784622610953349715890136060239421022216970366139189336822360...
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 + ...
DECIMAL EXPANSION TO 1000 DIGITS:
B = 0.89178462261095334971589013606023942102221697036613\
91893368223601276122378175444558396786463861763710\
57439093836139343959699895448987622772561974889829\
69662500641670749267412176492387283639777757763274\
25544373227852142261116843917982062828561973242641\
82725879555976060428390970218640637206146898948643\
76158809108390913335032108295905030664382411547224\
65652844918843557563559576104945928523599994449875\
54216008705234822642417410437080548464100874227218\
61650525099561200582641085028403673931750929494032\
47382019920912650558684222318629979407415580585052\
58521100916256823999312185479604796455256751507361\
67292078514305809228767193192555896703488660216859\
38438297427435171546623099960570301622830302948131\
42393878925766586388132889946469804516455360827301\
15060737460971066848430279446396669771028830058957\
09040428237475226018628287375514768624454713520927\
57806744194504585813229218682951533161650254564160\
40305474360667599580582080941206432281172119508572\
24718465451691587123672187602470833897922105839762...

Mathematica

digits = 120; B = NSum[(-1)^(n-1)/(2^n-1)^n, {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5]; RealDigits[B, 10, digits][[1]] (* Jean-François Alcover, Apr 25 2018 *)

Prog

(PARI) suminf(n=1, (-1)^(n-1)/(2^n-1)^n) \\ Michel Marcus, Apr 25 2018

Crossrefs

Cf. A303340 (binary), A300279.

Sequence in context: A132718 A154904 A155553 * A053463 A073636 A113521

Adjacent sequences: A302762 A302763 A302764 * A302766 A302767 A302768

Keyword

nonn,cons

Author

Paul D. Hanna, Apr 12 2018

Status

approved