A122579 Integer sequence arising from the nested radical R(m)=sqrt(4^0+sqrt(4^1+sqrt(....+sqrt(4^m)...) (see comment).

1, 1, 3, 15, 455, 53175, 52802967, 194817927511, 3103080851583191, 200041552904055799767, 52027813627693614618566103, 54341050032531342541499355961815

Offset

0,3

Comments

Let p(k)=prod(i=0,k,2^i+1) (A028361) then for any n>0 we get the asymptotic formula as m-->infty : 2-R(m)=a(0)*(1/p(m))+a(1)*(2/p(m+1))+...+a(n)*(2^n/p(m+n))+O(1/p(m+n+1)). In particular there is this unexpected relation between an infinite nested radical and an infinite product : limit m-->infty (2-R(m))*2^(m*(m+1)/2)=1/prod(i>=0,1+1/2^i)=0.29.... (A083864).

References

B. Cloitre, On an asymptotic formula for a nested radical, in preparation 2006

Links

Table of n, a(n) for n=0..11.

Formula

a(n) is asymptotic to c*4^(n*(n-1)/2) for c=0.041....

Crossrefs

Cf. A028361, A083864.

Sequence in context: A296940 A193119 A309069 * A138303 A338293 A267096

Adjacent sequences: A122576 A122577 A122578 * A122580 A122581 A122582

Keyword

nonn

Author

Benoit Cloitre, Oct 29 2006

Status

approved