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%I #32 Aug 06 2024 05:31:09
%S 1,6,3,2,8,4,3,0,1,8,0,4,3,7,8,6,2,8,7,4,1,6,1,5,9,4,7,5,0,6,1,0,5,0,
%T 4,4,3,4,0,6,6,2,2,7,5,1,8,4,1,1,0,5,6,0,8,6,8,2,4,2,1,8,0,7,6,8,6,1,
%U 1,1,2,2,8,3,8,9,1,1,0,6,0,0,1,2,0,9,7,0,6,2,6,4,9,6,7,9,4,5,3,1,2,3,5,1,1
%N Decimal expansion of C = Sum_{k>=0} 1/2^(2^k-1).
%C This constant has a nice continued fraction expansion (i.e. only 1 and 2 occur). C arises when looking for a sequence b(n) such that : b(1) = 0, b(n+1) is the smallest integer > b(n) such that the continued fraction for 1/2^b(1) + 1/2^b(2) + ... + 1/2^b(n+1) contains only 1's or 2's. Because b(n) = 2^n-1 and C = Sum_{k>=0} 1/2^b(k).
%H Harry J. Smith, <a href="/A076214/b076214.txt">Table of n, a(n) for n = 1..20000</a>
%H Boris Adamczewski, <a href="http://www.emis.de/journals/JIS/VOL16/Adamczewski/adam6.html">The Many Faces of the Kempner Number</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.15.
%H Michael Ian Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/PST.pdf">Property Enumerators and a Partial Sum Theorem</a>, 2011; <a href="https://citeseerx.ist.psu.edu/pdf/e503bec3c04c3f94cb267882724dd414e143141b">alternative link</a>.
%F Equals 2 * Sum_{k>=0} 1/2^(2^k) = 2 * A007404. - _Harry J. Smith_, May 09 2009
%F From _Amiram Eldar_, Mar 12 2024: (Start)
%F Equals 1 + 2 * A078585.
%F Equals 1 + Sum_{k>=1} floor(log_2(k))/2^k (Shamos, 2011, p. 8). (End)
%e 1.632843018043786287416159475061050443406622751841105608682421807686111...
%t Take[ RealDigits[ 2*NSum[1/2^2^k, {k, 0, Infinity}, WorkingPrecision -> 120]][[1]], 105] (* _Jean-François Alcover_, Nov 15 2011 *)
%o (PARI) default(realprecision, 20080); x=suminf(k=0, 1/2^(2^k)); x*=2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b076214.txt", n, " ", d)); \\ _Harry J. Smith_, May 09 2009
%Y Cf. A006466 (continued fraction), A007404, A078585.
%K cons,nonn,changed
%O 1,2
%A _Benoit Cloitre_, Nov 03 2002
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