

A094885


Decimal expansion of phi*e, where phi = (1 + sqrt(5))/2.


9



4, 3, 9, 8, 2, 7, 2, 3, 8, 9, 4, 4, 7, 9, 4, 6, 3, 9, 5, 9, 7, 1, 9, 8, 7, 0, 2, 9, 2, 9, 2, 8, 8, 5, 8, 6, 8, 7, 8, 6, 7, 4, 0, 4, 9, 7, 9, 7, 8, 8, 3, 4, 9, 1, 7, 0, 3, 8, 0, 9, 8, 0, 9, 0, 2, 1, 6, 4, 4, 4, 4, 3, 2, 1, 1, 6, 2, 0, 4, 4, 3, 0, 0, 3, 8, 5, 4, 6, 4, 3, 5, 2, 9, 2, 9, 4, 7, 2, 6
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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=(xd)*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

Cf. A001113, A001622, A104457, A094886.
Sequence in context: A222471 A180858 A263193 * A240199 A094728 A212001
Adjacent sequences: A094882 A094883 A094884 * A094886 A094887 A094888


KEYWORD

cons,nonn


AUTHOR

N. J. A. Sloane, Jun 15 2004


STATUS

approved



