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A273580
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Decimal expansion of the infinite nested radical sqrt(F_0 + sqrt(F_1 + sqrt(F_3 + ...))), where F_k are the Fermat numbers A000215.
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1
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2, 5, 2, 9, 5, 4, 3, 3, 2, 6, 2, 2, 0, 3, 9, 8, 4, 3, 0, 3, 1, 0, 3, 7, 9, 1, 2, 8, 8, 5, 9, 7, 5, 3, 3, 3, 5, 1, 9, 3, 5, 3, 7, 1, 2, 4, 4, 5, 9, 3, 8, 3, 4, 1, 7, 8, 6, 5, 7, 1, 8, 7, 1, 1, 3, 9, 6, 7, 3, 0, 9, 4, 6, 5, 4, 0, 4, 8, 7, 4, 8, 2, 5, 3, 1, 0, 3, 3, 5, 4, 4, 6, 0, 7, 2, 1, 5, 0, 0, 2, 3, 8, 9, 3, 3
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OFFSET
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1,1
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COMMENTS
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The convergence of this expression follows from Vijayaraghavan's theorem, for which it represents an extreme example.
Two PARI programs to compute this constant are listed below. The first one is a brute-force implementation of the definition and allows the computation of only 13 digits before exceeding current PARI capabilities. The second one implements the following 'trick' inspired by a comment in A094885: Let us try to compute first x = a/sqrt(2). We have x = (1/sqrt(2))sqrt(3+ sqrt(5+ sqrt(17+ ... ))) = sqrt(3/2+ (1/2)sqrt(5+ sqrt(17+ ... ))) = sqrt(3/2+ sqrt(5/4+ (1/4)sqrt(17+ ... ))) = sqrt(3/2+ sqrt(5/4+ sqrt(17/16+ ... ))) = sqrt(c_0+sqrt(c_1+sqrt(c_3+...))), where c_n = (2^(2^n)+1)/2^(2^n) = 1+d_n, with d_n = 2^(-2^n). This nested radical is easy to manage to any precision. However, evaluating it up to N terms, its convergence with increasing N is no better than that of the original algorithm. To speed it up, one must notice that, since the c_n converge rapidly to 1, and since the nested radical sqrt(1+sqrt(1+...)) evaluates to the golden ratio phi (A001622), the latter is the natural best stand-in for the neglected part (terms from N+1 to infinity). With this modification, i.e., 'seeding' the iterations with phi instead of 0, the convergence becomes extremely fast (the number of valid digits more than doubles upon incrementing N by 1).
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LINKS
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FORMULA
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Equals sqrt(2)*sqrt(1+1/2+sqrt(1+1/4+sqrt(1+1/16+sqrt(1+1/256+ ... )))).
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EXAMPLE
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2.5295433262203984303103791288597533351935371244593834178657187113967...
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PROG
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(PARI) /* This function crashes PARI beyond N=28: */
s(N)={my(r=0.0); for(k=1, N, r=sqrt(2^(2.0^(N-k))+1+r)); return(r)}
/* N is the number of terms to include in the evaluation. It turns out that the starting digits s(28) shares with s(27) are only 13 */
(PARI) /* This alternative can easily generate millions of digits: */
d=vector(30); d[1]=0.5; for(n=2, #d, d[n]=d[n-1]^2);
S(N)={my(r=(1+sqrt(5))/2); for(k=1, N, r=sqrt(1+d[N-k+1]+r)); return(r*sqrt(2))}
/* S(12) exceeds 1200 stable digits, S(20) goes over 150000. For the b-file, the first 2000 digits of S(13) were used, computed with the realprecision of 2100 digits */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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