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A007400
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Continued fraction for Sum_{n>=0} 1/2^(2^n) = 0.8164215090218931...
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30
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0, 1, 4, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 6, 4
(list;
graph;
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listen;
history;
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OFFSET
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0,3
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REFERENCES
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M. Kmošek, Rozwinieçie Niektórych Liczb Niewymiernych na Ulamki Lancuchowe (Continued Fraction Expansion of Some Irrational Numbers), Master's thesis, Uniwersytet Warszawski, 1979.
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LINKS
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FORMULA
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a(0)=0, a(1)=1, a(2)=4; for n > 2:
a(8k) = a(8k+3) = 2;
a(8k+4) = a(8k+7) = a(16k+5) = a(16k+14) = 4;
a(16k+6) = a(16k+13) = 6;
a(8k+1) = a(4k+1);
a(8k+2) = a(4k+2). (End)
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EXAMPLE
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0.816421509021893143708079737... = 0 + 1/(1 + 1/(4 + 1/(2 + 1/(4 + ...))))
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MAPLE
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a:= proc(n) option remember; local n8, n16;
n8:= n mod 8;
if n8 = 0 or n8 = 3 then return 2
elif n8 = 4 or n8 = 7 then return 4
elif n8 = 1 then return procname((n+1)/2)
elif n8 = 2 then return procname((n+2)/2)
fi;
n16:= n mod 16;
if n16 = 5 or n16 = 14 then return 4
elif n16 = 6 or n16 = 13 then return 6
fi
end proc:
a(0):= 0: a(1):= 1: a(2):= 4:
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MATHEMATICA
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a[n_] := a[n] = Which[n < 3, {0, 1, 4}[[n+1]], Mod[n, 8] == 1, a[(n+1)/2], Mod[n, 8] == 2, a[(n+2)/2], True, {2, 0, 0, 2, 4, 4, 6, 4, 2, 0, 0, 2, 4, 6, 4, 4}[[Mod[n, 16]+1]]]; Table[a[n], {n, 0, 98}] (* Jean-François Alcover, Nov 29 2013, after Ralf Stephan *)
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PROG
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(PARI) a(n)=if(n<3, [0, 1, 4][n+1], if(n%8==1, a((n+1)/2), if(n%8==2, a((n+2)/2), [2, 0, 0, 2, 4, 4, 6, 4, 2, 0, 0, 2, 4, 6, 4, 4][(n%16)+1]))) /* Ralf Stephan */
(PARI) a(n)=contfrac(suminf(n=0, 1/2^(2^n)))[n+1]
(PARI) { allocatemem(932245000); default(realprecision, 26000); x=suminf(n=0, 1/2^(2^n)); x=contfrac(x); for (n=1, 20001, write("b007400.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 07 2009
(Scheme) (define (A007400 n) (cond ((<= n 1) n) ((= 2 n) 4) (else (case (modulo n 8) ((0 3) 2) ((4 7) 4) ((1) (A007400 (/ (+ 1 n) 2))) ((2) (A007400 (/ (+ 2 n) 2))) (else (case (modulo n 16) ((5 14) 4) ((6 13) 6))))))) ;; (After Ralf Stephan's recurrence) - Antti Karttunen, Aug 12 2017
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CROSSREFS
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KEYWORD
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nonn,cofr,easy
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AUTHOR
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STATUS
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approved
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