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Decimal expansion of Sum_{n>=0} 1/4^(2^n).
12

%I #24 Nov 12 2020 05:17:10

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

%T 2,1,7,0,3,3,1,1,3,7,5,9,2,0,5,5,2,8,0,4,3,4,1,2,1,0,9,0,3,8,4,3,0,5,

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

%N Decimal expansion of Sum_{n>=0} 1/4^(2^n).

%H Harry J. Smith, <a href="/A078585/b078585.txt">Table of n, a(n) for n = 0..20000</a>

%H Aubrey J. Kempner, <a href="https://doi.org/10.1090/S0002-9947-1916-1501054-4">On Transcendental Numbers</a>, Transactions of the American Mathematical Society, volume 17, number 4, October 1916, pages 476-482.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals -Sum_{k>=1} mu(2*k)/(4^k - 1), where mu is the Möbius function (A008683). - _Amiram Eldar_, Jul 12 2020

%F Equals A007404 - 1/2. - _Kevin Ryde_, Nov 11 2020

%e 0.316421509021893143708079737530525221703311375920552804341210903843055...

%t RealDigits[ N[ Sum[1/4^(2^n), {n, 0, Infinity}], 110]][[1]]

%o (PARI) { default(realprecision, 20080); x=suminf(n=0, 1/4^(2^n)); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b078585.txt", n, " ", d)); } \\ _Harry J. Smith_, May 11 2009

%Y Continued fraction is given in A006464.

%Y Similar sums: A007404, A078885, A078886, A078887, A078888, A078889, A078890, A036987.

%K cons,nonn

%O 0,1

%A _Robert G. Wilson v_, Dec 01 2002