OFFSET

1,1

COMMENTS

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).

LINKS

Stanislav Sykora, Table of n, a(n) for n = 1..2000

FORMULA

Equals sqrt(2)*sqrt(1+1/2+sqrt(1+1/4+sqrt(1+1/16+sqrt(1+1/256+ ... )))).

EXAMPLE

2.5295433262203984303103791288597533351935371244593834178657187113967...

PROG

(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 */

CROSSREFS

KEYWORD

nonn,cons

AUTHOR

Stanislav Sykora, May 25 2016

STATUS

approved