A083864 -id:A083864 - cp's OEIS frontend

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

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Benoit Cloitre, Oct 29 2006

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Let p(k) = Product_{i=0..k} (2^i+1) (A028361) then for any n>0 we get the asymptotic formula: lim_{m->oo} 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: lim_{m->oo} (2-R(m))*2^(m*(m+1)/2) = 1/Product_{i>=0} (1+1/2^i) = 0.209... (A083864).

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

Cf. A028361, A083864.

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

cp's OEIS Frontend

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

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