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Decimal expansion of the sum of reciprocals of repunit numbers base 4, Sum_{k>=1} 3/(4^k - 1).
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%I #28 Jun 03 2020 07:13:15

%S 1,2,6,3,2,9,3,0,5,8,1,0,0,2,7,1,3,3,1,8,8,7,9,7,2,6,6,3,9,0,3,1,3,9,

%T 1,4,6,8,8,4,3,2,4,0,0,8,9,7,2,3,4,6,2,1,3,8,1,7,7,6,2,3,9,0,1,3,8,3,

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

%N Decimal expansion of the sum of reciprocals of repunit numbers base 4, Sum_{k>=1} 3/(4^k - 1).

%C The sums of reciprocal repunit numbers are related to the Lambert series. A special case is the sum of repunit numbers in base 2, which is known as the Erdős-Borwein constant (A065442).

%H N. Kurakowa and M. Wakyama, <a href="https://doi.org/10.1090/S0002-9939-03-07025-4">On q-analogues of the Euler Constant and Lerch's limit formula</a>, Proc. AMS 132 (4) (2003) 935, constant gamma(4).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Erdos-BorweinConstant.html">Erdős-Borwein Constant</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertSeries.html">Lambert Series</a>

%F Equals 3*L(1/4) = 3 * A248721, where L is the Lambert series.

%F Equals 3 * Sum_{k>=1} x^(k^2)*(1+x^k)/(1-x^k) where x = 1/4.

%e 1.263293058100271331887972663903139146884324008972346213817762390...

%p evalf[130](sum(3/(4^k-1),k=1..infinity)); # _Muniru A Asiru_, Dec 20 2018

%t RealDigits[Sum[3/(4^k-1), {k, 1, Infinity}], 10, 120][[1]] (* _Amiram Eldar_, Nov 21 2018 *)

%o (PARI) suminf(k=1, 3/(4^k-1)) \\ _Michel Marcus_, Nov 20 2018

%Y Cf. A002450, A065442 (base 2), A321872 (base 3).

%K nonn,cons

%O 1,2

%A _A.H.M. Smeets_, Nov 20 2018