A094214 Decimal expansion of 1/phi = phi - 1.

6, 1, 8, 0, 3, 3, 9, 8, 8, 7, 4, 9, 8, 9, 4, 8, 4, 8, 2, 0, 4, 5, 8, 6, 8, 3, 4, 3, 6, 5, 6, 3, 8, 1, 1, 7, 7, 2, 0, 3, 0, 9, 1, 7, 9, 8, 0, 5, 7, 6, 2, 8, 6, 2, 1, 3, 5, 4, 4, 8, 6, 2, 2, 7, 0, 5, 2, 6, 0, 4, 6, 2, 8, 1, 8, 9, 0, 2, 4, 4, 9, 7, 0, 7, 2, 0, 7, 2, 0, 4, 1, 8, 9, 3, 9, 1, 1, 3, 7, 4, 8, 4

Offset: 0

Author

Cino Hilliard, May 27 2004

Keywords

• cons
• nonn
• easy

Comments

Edge length of a regular decagon with unit circumradius. - Stanislav Sykora, May 07 2014

The value a+0i is the only invariant point of the complex-plane endomorphism M(z)=sqrt(2-sqrt(2+z)), and also its unique attractor, with the iterations converging exponentially from any starting complex value. Hence the infinite radical formula. - Stanislav Sykora, Apr 29 2016

With a minus sign, this constant is called beta and shares many identities with phi = A001622 (also called alpha); e.g., beta * phi = -1, Lucas numbers L(n) = A000032(n) = phi^n + beta^n. - Andrés Ventas, Apr 23 2022

Examples

			0.6180339887498948482045868343656381177203091798057628621354486227052604628...
		

References

• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 137-138, 178-180, 257.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• Aziz El Kacimi, Des triangles dorés, Images des Mathématiques, CNRS, 2016 (in French).
• Eric Weisstein's World of Mathematics, Decagon.
• Eric Weisstein's World of Mathematics, Golden Ratio Conjugate.
• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A001622, A104457, A000032.

Programs

Mathematica

RealDigits[N[GoldenRatio-1,200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011*)

PARI

default(realprecision, 20080); x=(sqrt(5)-1)/2; d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b094214.txt", n, " ", d));  \\ Harry J. Smith, Apr 19 2009

PARI

(sqrt(5)-1)/2 \\ Michel Marcus, Mar 21 2016

PARI

a(n) = floor( 10^(n+1)*(quadgen(5)-1)%10);
alist(len) = digits(floor((quadgen(5)-1)*10^len)); \\ Chittaranjan Pardeshi, May 31 2022

Formula

Equals A001622 -1 .

Equals sqrt(2-sqrt(2+sqrt(2-sqrt(2+ ...)))). - Stanislav Sykora, Apr 29 2016

From Christian Katzmann, Mar 19 2018: (Start)

Equals Sum_{n>=0} (15*(2*n)!-8*n!^2)/(2*n!^2*3^(2*n+2)).

Equals -1/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)

Equals i^(4/5) + i^(-4/5). - Gary W. Adamson, Feb 05 2022

From Andrés Ventas, Apr 23 2022: (Start)

Equals (sqrt(5)-1)/2.

Equals 2*sin(Pi/10). (End)

Equals tan(arctan(2)/2). - Amiram Eldar, Jun 29 2023

Positive solution y to y = Integral_{0..1} x^y dx. - Andrea Pinos, Jun 24 2024

Extensions

Edited by Eric W. Weisstein, Apr 17 2006