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%I M0049 #34 Aug 30 2017 17:50:19
%S 1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,2,1,2,1,1,1,1,2,1,1,1,1,
%T 1,1,1,2,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,2,1,1,1,1,2,1,1,1,
%U 1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,2,1,2
%N Continued fraction expansion of C = 2*Sum_{n>=0} 1/2^(2^n).
%C C arises when looking for a sequence b(n) such that b(1)=0 and 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. It arises because b(n) = 2^n - 1 and C = Sum_{k>=0} 1/2^b(k). - _Benoit Cloitre_, Nov 03 2002
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Harry J. Smith, <a href="/A006466/b006466.txt">Table of n, a(n) for n = 0..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), #13.2.15.
%H J. Shallit, <a href="/A006463/a006463.pdf">Letter to N. J. A. Sloane with attachment, Aug. 1979</a>
%H Jeffrey Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/scf.ps">Simple continued fractions for some irrational numbers</a>. J. Number Theory 11 (1979), no. 2, 209-217 <a href="http://dx.doi.org/10.1016/0022-314X(79)90040-4">[DOI]</a>
%F Recurrence: a(5n) = a(5n+1) = a(2) = a(5n+3) = a(20n+14) = a(40n+9) = 1, a(20n+4) = a(40n+29) = 2, a(5n+2) = 3 - a(5n-1), a(20n+19) = a(10n+9). - _Ralf Stephan_, May 17 2005
%e 1.632843018043786287416159475... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, May 09 2009
%o (PARI) { allocatemem(932245000); default(realprecision, 10000); x=suminf(n=0, 1/2^(2^n)); x=contfrac(2*x); for (n=1, 20001, write("b006466.txt", n-1, " ", x[n])); } \\ _Harry J. Smith_, May 09 2009
%Y Cf. A076214 = Decimal expansion. - _Harry J. Smith_, May 09 2009
%K nonn,cofr
%O 0,5
%A _N. J. A. Sloane_
%E Better description and more terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 19 2001