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A182007 Decimal expansion of 2*sin(Pi/5); the 'associate' of the golden ratio. 9
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 (list; constant; graph; refs; listen; history; text; internal format)
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((5-sqrt(5))/2), which is the eccentricity of a golden ellipse, that is, an ellipse inscribed in a golden rectangle. - Jean-Fran├žois 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

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); rms(c,phi) = sqrt((c^2+phi^2)/2) = sqrt(2).

EXAMPLE

1.1755705045849462583374119...

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

CROSSREFS

Cf. A001622, 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

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Last modified April 30 08:33 EDT 2017. Contains 285645 sequences.