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A019863 Decimal expansion of sine of 54 degrees. 12
8, 0, 9, 0, 1, 6, 9, 9, 4, 3, 7, 4, 9, 4, 7, 4, 2, 4, 1, 0, 2, 2, 9, 3, 4, 1, 7, 1, 8, 2, 8, 1, 9, 0, 5, 8, 8, 6, 0, 1, 5, 4, 5, 8, 9, 9, 0, 2, 8, 8, 1, 4, 3, 1, 0, 6, 7, 7, 2, 4, 3, 1, 1, 3, 5, 2, 6, 3, 0, 2, 3, 1, 4, 0, 9, 4, 5, 1, 2, 2, 4, 8, 5, 3, 6, 0, 3, 6, 0, 2, 0, 9, 4, 6, 9, 5, 5, 6, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Midsphere radius of regular icosahedron with unit edges.

Also half of the golden ratio (A001622). - Stanislav Sykora, Jan 30 2014

Andris Ambainis (see Aaronson link) observes that combining the results of Barak-Hardt-Haviv-Rao with Dinur-Steurer yields the maximal probability of winning n parallel repetitions of a classical CHSH game (see A201488) asymptotic to this constant to the power of n, an improvement on the naive probability of (3/4)^n. (All the random bits are received upfront but the players cannot communicate or share an entangled state.) - Charles R Greathouse IV, May 15 2014

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Scott Aaronson, The NEW Ten Most Annoying Questions in Quantum Computing (2014)

Boaz Barak, Moritz Hardt, Ishay Haviv, and Anup Rao, Rounding Parallel Repetitions of Unique Games (2008)

Irit Dinur and David Steurer, Analytical approach to parallel repetition (2013)

Wikipedia, Exact trigonometric constants

Wikipedia, Platonic solid

FORMULA

Equals [1+sqrt(5)]/4 = cos(Pi/5) = sin(3Pi/10). - R. J. Mathar, Jun 18 2006

Equals 2F1(4/5,1/5;1/2;3/4) / 2 = A019827 + 1/2. - R. J. Mathar, Oct 27 2008

Equals A001622 / 2. - Stanislav Sykora, Jan 30 2014

phi / 2 = (i^(2/5) + i^(-2/5)) / 2 = i^(2/5) - (sin(Pi/5))*i = i^(-2/5) + (sin(Pi/5))*i = i^(2/5) - (cos(3*Pi/10))*i = i^(-2/5) + (cos(3*Pi/10))*i. - Jaroslav Krizek, Feb 03 2014

EXAMPLE

0.80901699437494742410229341718281905886015458990288143106772431135263...

MAPLE

Digits:=100; evalf((1+sqrt(5))/4); # Wesley Ivan Hurt, Mar 27 2014

MATHEMATICA

RealDigits[(1 + Sqrt[5])/4, 10, 111] (* Robert G. Wilson v *)

PROG

(PARI) (1+sqrt(5))/4 \\ Charles R Greathouse IV, Jan 16 2012

CROSSREFS

Cf. A019827, A001622.

Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A239798 (dodecahedron).

Sequence in context: A076350 A197617 A005076 * A243456 A198820 A138285

Adjacent sequences:  A019860 A019861 A019862 * A019864 A019865 A019866

KEYWORD

nonn,cons,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 2 08:26 EDT 2014. Contains 246339 sequences.