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A094214 Decimal expansion of 1/phi = phi - 1. 66
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 (list; constant; graph; refs; listen; history; text; internal format)
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
LINKS
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.
FORMULA
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
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
Sequence in context: A143019 A337369 A156921 * A001622 A186099 A021622
KEYWORD
cons,nonn,easy
AUTHOR
Cino Hilliard, May 27 2004
EXTENSIONS
Edited by Eric W. Weisstein, Apr 17 2006
STATUS
approved

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Last modified June 23 16:08 EDT 2024. Contains 373651 sequences. (Running on oeis4.)