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%I M1312 #40 Aug 29 2022 13:04:20
%S 0,2,5,3,3,1,3,5,3,1,5,3,1,3,3,5,3,1,5,3,3,1,3,5,1,3,5,3,1,3,3,5,3,1,
%T 5,3,3,1,3,5,3,1,5,3,1,3,3,5,1,3,5,3,3,1,3,5,1,3,5,3,1,3,3,5,3,1,5,3,
%U 3,1,3,5,3,1,5,3,1,3,3,5,3,1,5,3,3,1,3,5,1,3,5,3,1,3,3,5,1,3,5,3,3,1,3,5,3
%N Continued fraction for Sum_{k>=0} 1/3^(2^k).
%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="/A004200/b004200.txt">Table of n, a(n) for n = 0..20000</a>
%H Jeffrey O. Shallit, <a href="/A006463/a006463.pdf">Letter to N. J. A. Sloane with attachment, Aug. 1979</a>
%H Jeffrey O. 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.
%H Jeffrey O. Shallit, <a href="http://dx.doi.org/10.1016/0022-314X(79)90040-4">Simple continued fractions for some irrational numbers</a>, J. Number Theory 11 (1979), no. 2, 209-217.
%H Jeffrey O. Shallit, <a href="https://doi.org/10.1016/0022-314X(82)90047-6">Simple Continued Fractions for Some Irrational Numbers II</a>, J. Number Theory 14 (1982), 228-231.
%F Recurrence: a(0)=0, a(1)=2, a(2)=5, a(16n+5)=a(16n+12)=a(32n+9)=a(32n+24)=1, a(8n+3)=a(8n+6)=a(16n+4)=a(16n+13)=a(32n+8)=a(32n+25)=3, a(8n+2)=a(8n+7)=5, a(16n)=a(8n), a(16n+1)=a(8n+1). - _Ralf Stephan_, May 17 2005
%e 0.456942562477639661115491826... = 0 + 1/(2 + 1/(5 + 1/(3 + 1/(3 + ...)))).
%p u := 3: v := 7: Buv := [u,1,[0,u-1,u+1]]: for k from 2 to v do n := nops(Buv[3]): Buv := [u,Buv[2]+1,[seq(Buv[3][i],i=1..n-1),Buv[3][n]+1,Buv[3][n]-1,seq(Buv[3][n-i],i=1..n-2)]] od: seq(Buv[3][i],i=1..2^v);# first 2^v terms of A004200 # Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Dec 02 2002
%t ContinuedFraction[ NSum[1/3^(2^n), {n, 0, Infinity}, WorkingPrecision -> 105], 105] (* _Jean-François Alcover_, Jul 18 2011 *)
%o (PARI) { allocatemem(932245000); default(realprecision, 20000); x=suminf(n=0, 1/3^(2^n)); x=contfrac(x); for (n=1, 20001, write("b004200.txt", n-1, " ", x[n])); } \\ _Harry J. Smith_, May 10 2009
%Y Cf. A007400, A078885 (decimal expansion).
%K nonn,cofr
%O 0,2
%A _N. J. A. Sloane_
%E Better description and more terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 19 2001
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