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

0,1

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

The relative asymptotic density of the even terms in the sequence of Fibbinary numbers (A003714). - Amiram Eldar, Jan 06 2026

References

Ralph P. Grimaldi, Fibonacci and Catalan Numbers: An Introduction, 2012. See Exercises 13 and 22 at p. 59.

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).

V. E. Hoggatt and D. A. Lind, The Heights of Fibonacci Polynomials and an Associated Function, Fibonacci Quarterly, Vol. 5, No. 2 (April, 1967), pp. 141-152. See Theorem 2 at page 143.

Eric Weisstein's World of Mathematics, Decagon.

Eric Weisstein's World of Mathematics, Golden Ratio Conjugate.

Index entries for algebraic numbers, degree 2.

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

Equals lim_{n->oo} 2*n/(n + 1 + sqrt(5*n^2 - 2*n + 1)) [Hoggatt and Lind, 1967]. - Stefano Spezia, Nov 16 2025

Equals Sum_{k>=0} binomial(3*k+1, k)*2^(-3*k-1)/(3*k+1). - Stefano Spezia, Dec 23 2025

Example

0.6180339887498948482045868343656381177203091798057628621354486227052604628...

Mathematica

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

Prog

(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

Crossrefs

Cf. A000032, A001622, A003714, A104457.

Sequence in context: A143019 A337369 A156921 * A001622 A385045 A186099

Adjacent sequences: A094211 A094212 A094213 * A094215 A094216 A094217

Keyword

cons,nonn,easy

Author

Cino Hilliard, May 27 2004

Extensions

Edited by Eric W. Weisstein, Apr 17 2006

Status

approved