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%I #16 Apr 26 2018 09:57:48
%S 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,
%T 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,
%U 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
%N Decimal expansion of constant: B = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.
%H Paul D. Hanna, <a href="/A302765/b302765.txt">Table of n, a(n) for n = 0..1030</a>
%F This constant may be defined by the following expressions.
%F (1) B = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.
%F (2) B = Sum_{n>=1} (2^n - 1)^(n-1) / 2^(n^2).
%F (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.
%e Constant B = 0.891784622610953349715890136060239421022216970366139189336822360...
%e This constant equals the sum of the following infinite series.
%e (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 + ...
%e Also,
%e (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 + ...
%e Expressed in terms of powers of 1/2, we have
%e (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 + ...
%e DECIMAL EXPANSION TO 1000 DIGITS:
%e B = 0.89178462261095334971589013606023942102221697036613\
%e 91893368223601276122378175444558396786463861763710\
%e 57439093836139343959699895448987622772561974889829\
%e 69662500641670749267412176492387283639777757763274\
%e 25544373227852142261116843917982062828561973242641\
%e 82725879555976060428390970218640637206146898948643\
%e 76158809108390913335032108295905030664382411547224\
%e 65652844918843557563559576104945928523599994449875\
%e 54216008705234822642417410437080548464100874227218\
%e 61650525099561200582641085028403673931750929494032\
%e 47382019920912650558684222318629979407415580585052\
%e 58521100916256823999312185479604796455256751507361\
%e 67292078514305809228767193192555896703488660216859\
%e 38438297427435171546623099960570301622830302948131\
%e 42393878925766586388132889946469804516455360827301\
%e 15060737460971066848430279446396669771028830058957\
%e 09040428237475226018628287375514768624454713520927\
%e 57806744194504585813229218682951533161650254564160\
%e 40305474360667599580582080941206432281172119508572\
%e 24718465451691587123672187602470833897922105839762...
%t 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 *)
%o (PARI) suminf(n=1, (-1)^(n-1)/(2^n-1)^n) \\ _Michel Marcus_, Apr 25 2018
%Y Cf. A303340 (binary), A300279.
%K nonn,cons
%O 0,1
%A _Paul D. Hanna_, Apr 12 2018
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