A323482 - OEIS

A323482

Decimal expansion of 1/2 + 1/3 + 1/5 + ... + 1/(2^n + 1) + ...

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

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OFFSET

1,2

LINKS

Jinyuan Wang, Table of n, a(n) for n = 1..1000

FORMULA

From Amiram Eldar, Jun 30 2020: (Start)

Equals 1/2 + Sum_{k>=1} (-1)^(k+1)/(2^k-1)

Equals Sum_{k>=1} (mu(k) - (-1)^k)/(2^k-1), where mu is the Möbius function (A008683).

Equals (1 + A179951)/2. (End)

Equals (1/2) + Sum_{n >= 1} (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2). The first 18 terms of the series gives the constant correct to more than 100 decimal places. - Peter Bala, Jan 28 2022

EXAMPLE

1.2644997803484442091913...

MAPLE

evalf((1/2) + add( (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2), n = 1..18), 100); # Peter Bala, Jan 28 2022

MATHEMATICA

s = Sum[1/(2^k + 1), {k, 0, Infinity}]

r = N[Re[s], 200]

RealDigits[r][[1]]

PROG

(PARI) suminf(k=0, 1/(2^k+1)) \\ Michel Marcus, Jan 15 2019

CROSSREFS

Cf. A000051, A065442, A008683, A179951.