

A182007


Decimal expansion of 2*sin(Pi/5); the 'associate' of the golden ratio.


14



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

1,3


COMMENTS

Golden ratio phi is the real part of 2*exp(i*Pi/5), while this constant c is the corresponding imaginary part. It is handy, for example, in simplifying metric expressions for Platonic solids (particularly for regular icosahedron and dodecahedron).
Note that c^2+A001622^2 = 4; c*A001622 = A188593 = 2*A019881; c = 2*A019845.
Equals sqrt((5sqrt(5))/2), which is the eccentricity of a golden ellipse, that is, an ellipse inscribed in a golden rectangle.  JeanFrançois Alcover, May 21 2013
Edge length of a regular pentagon with unit circumradius.  Stanislav Sykora, May 07 2014
This is a constructible number (see A003401 for more details). Moreover, since phi is also constructible, (2^k)*exp(i*Pi/5), for any integer k, is a constructible complex number.  Stanislav Sykora, May 02 2016
rms(c, phi) := sqrt((c^2+phi^2)/2) = sqrt(2) = A002193.


LINKS

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pentagon
Wikipedia, Platonic solid


FORMULA

c = 2*sin(Pi/5) = sqrt(3phi).


EXAMPLE

1.1755705045849462583374119...


MAPLE

evalf(2*sin(Pi/5), 100); # Muniru A Asiru, Nov 02 2018


MATHEMATICA

RealDigits[2*Sin[Pi/5], 10, 120][[1]] (* Harvey P. Dale, Sep 29 2012 *)


PROG

(PARI) 2*sin(Pi/5) \\ Stanislav Sykora, May 02 2016
(MAGMA) SetDefaultRealField(RealField(100)); R:= RealField(); 2*Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018


CROSSREFS

Cf. A001622, A002193, A003401, A019881, A019845, A102769, A131595, A188593.
Sequence in context: A070273 A158244 A226580 * A247320 A179294 A259679
Adjacent sequences: A182004 A182005 A182006 * A182008 A182009 A182010


KEYWORD

nonn,cons,easy


AUTHOR

Stanislav Sykora, Apr 06 2012


STATUS

approved



