OFFSET
1,1
COMMENTS
Matches the value of the infinite nested radical corresponding to the sequence {e^(2^n), n=1,2,3,...}, i.e., a = sqrt(e^2+sqrt(e^4+...)), which converges by Vijayaraghavan's theorem. Proof: write the golden ratio as phi = sqrt(1+ sqrt(1+ sqrt(1+...))). Then e*phi = e*sqrt(1+ sqrt(1+ sqrt(1+ ...))) = sqrt(e^2+ e^2*sqrt(1+ sqrt(1+ ...))) = sqrt(e^2+ sqrt(e^4+ e^4*sqrt(1+ ...))) = ... = a. Evidently, the 'e' could stand for any constant, not just e; for example phi itself as in A104457, or Pi as in A094886. - Stanislav Sykora, May 24 2016
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Nested Radical
EXAMPLE
4.398272389447946...
MATHEMATICA
First@ RealDigits[N[GoldenRatio E, 120]] (* Michael De Vlieger, May 24 2016 *)
PROG
(PARI) default(realprecision, 20080); phi=(1+sqrt(5))/2; x=phi*exp(1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b094885.txt", n, " ", d)); \\ Harry J. Smith, Apr 27 2009
(PARI) exp(1)*(1+sqrt(5))/2 \\ Michel Marcus, May 25 2016
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
N. J. A. Sloane, Jun 15 2004
STATUS
approved