A303560 Decimal expansion of constant A = Sum_{n>=1} 1 / (2^n - 1)^n.

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

Offset

1,4

Links

Table of n, a(n) for n=1..120.

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

Constant 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 + ...
DECIMAL EXPANSION TO 1000 DIGITS:
A = 1.11404635103800304961499423620017724756514316555838\
90230651114540148186855492164961058034546715309853\
47877270372301842177485420929460338909110702521744\
10383049196253371844115566456211414378684927895066\
60974873819605352670009454376709247947228660654797\
93238935770616752469239881642090329202510251771440\
45431299113929370687739805515426044704234082381940\
40977172853717815297745712948744536180513363052564\
03854647492812806063479313722184475876462500578835\
52045304926771113275210795841642087115096877536105\
78958100347787242164699010158545554930990338272655\
43040184293715344496344121360156193744995971261933\
73889789233802551892919983961109429243561889220300\
14247648527081291849257683339708248465152426049641\
37604659963590944969427064766226893075229683791387\
71510378240823470806036647280652258639308495696013\
02313861338203801779994141266165775176960964298159\
94404826055321122142831544652111697011832111972164\
78072293762844331943953407090067036379926709332730\
54952123380671191947076262400344800454366565708630...

Crossrefs

Cf. A303561 (binary), A302765, A143862.

Sequence in context: A088375 A217220 A361620 * A341481 A135011 A184907

Adjacent sequences: A303557 A303558 A303559 * A303561 A303562 A303563

Keyword

nonn,cons

Author

Paul D. Hanna, Apr 26 2018

Status

approved