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%I M2512 #29 Aug 30 2017 17:49:38
%S 0,3,6,4,4,2,4,6,4,2,6,4,2,4,4,6,4,2,6,4,4,2,4,6,2,4,6,4,2,4,4,6,4,2,
%T 6,4,4,2,4,6,4,2,6,4,2,4,4,6,2,4,6,4,4,2,4,6,2,4,6,4,2,4,4,6,4,2,6,4,
%U 4,2,4,6,4,2,6,4,2,4,4,6,4,2,6,4,4,2,4,6,2,4,6,4,2,4,4,6,2,4,6,4,4,2,4,6,4
%N Continued fraction for Sum_{n>=0} 1/4^(2^n).
%C a(n)=A004200(n) if n=0; A004200(n)+1 if n>0 (according to case u=3, b=1 of Theorem 5 (of the reference) which states that: if B(u,infinity) = Sum_{n>=0} 1/u^(2^n) = [a0, a1, a2, ...] then B(u + b,infinity) = [a0, a1+b, a2+b, a3+b,... ] (u >= 3, b >= 0)).
%C The sum is equal to 0.316421509021893143708079...= A078585.
%C After computing the first 10^5 terms and dropping the first two (0 & 3), only the numbers 2, 4 & 6 occur. Further I found no two 0's in a row and no three 2's or three 1's in a row. - _Robert G. Wilson v_, Dec 01 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="/A006464/b006464.txt">Table of n, a(n) for n = 0..20000</a>
%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>
%e 0.316421509021893143708079737... = 0 + 1/(3 + 1/(6 + 1/(4 + 1/(4 + ...)))). - _Harry J. Smith_, May 11 2009
%p u := 4: 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 A006464, Antonio G. Astudillo (aft_astudillo(AT)hotmail.com), Dec 02 2002
%t ContinuedFraction[ N[ Sum[1/4^(2^n), {n, 0, Infinity}], 1000]]
%o (PARI) { allocatemem(932245000); default(realprecision, 25000); x=suminf(n=0, 1/4^(2^n)); x=contfrac(x); for (n=1, 20001, write("b006464.txt", n-1, " ", x[n])); } \\ _Harry J. Smith_, May 11 2009
%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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