%I #39 Jan 30 2022 15:57:33
%S 4,2,1,0,9,7,6,8,6,0,3,3,4,2,3,7,7,7,2,9,5,9,9,0,8,8,7,9,6,7,7,1,3,0,
%T 4,8,9,6,1,4,4,1,3,3,6,3,2,4,1,1,5,4,0,4,6,0,5,9,2,0,7,9,6,7,1,2,7,7,
%U 1,3,7,0,4,8,8,7,3,9,8,0,2,7,5,1,9,0,3,6,8,4,7,5,8,6,5,0,7,9,5,3,9,2,8,4,5
%N Decimal expansion of Sum_{k>=1} 1/(4^k - 1).
%H G. C. Greubel, <a href="/A248721/b248721.txt">Table of n, a(n) for n = 0..10000</a>
%F Equals Sum_{k>=1} x^(k^2)*(1+x^k)/(1-x^k) where x = 1/4 (the Lambert series evaluated at 1/4). - _Joerg Arndt_, Jun 03 2020
%F Equals Sum_{k>=1} d(k)/4^k, where d(k) is the number of divisors of k (A000005). - _Amiram Eldar_, Jun 22 2020
%e 0.4210976860334237772959908879677130489614413363241154046059207967127713704887...
%p evalf(sum(1/(4^k-1), k=1..infinity),120) # _Vaclav Kotesovec_, Oct 18 2014
%p # second program with faster converging series after _Joerg Arndt_
%p evalf( add( (1/4)^(n^2)*(1 + 2/(4^n - 1)), n = 1..13), 105); # _Peter Bala_, Jan 30 2022
%t x = 1/4; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of _Amarnath Murthy_, see A073668 *)
%o (PARI) suminf(k=1, 1/(4^k-1)) \\ _Michel Marcus_, Oct 18 2014
%Y Cf. A000005, A065442, A073668, A214369, A248722, A248723, A248724, A248725, A248726.
%K nonn,cons
%O 0,1
%A _Robert G. Wilson v_, Oct 12 2014