A094885 Decimal expansion of phi*e, where phi = (1 + sqrt(5))/2.

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

Offset: 1

Author

N. J. A. Sloane, Jun 15 2004

Keywords

• cons
• nonn

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

Examples

			4.398272389447946...
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• Eric Weisstein's World of Mathematics, Nested Radical
• Index entries for transcendental numbers

Crossrefs

Cf. A001113, A001622, A104457, A094886.

Programs

Mathematica

First@ RealDigits[N[GoldenRatio E, 120]] (* Michael De Vlieger, May 24 2016 *)

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