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%I A019863 #104 Nov 28 2024 07:46:46
%S A019863 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,
%T A019863 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,
%U A019863 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
%N A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).
%C A019863 Midsphere radius of regular icosahedron with unit edges.
%C A019863 Also half of the golden ratio (A001622). - _Stanislav Sykora_, Jan 30 2014
%C A019863 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
%C A019863 This is the height h of the isosceles triangle in a regular pentagon, in length units of the circumscribing radius, formed by a side as base and two adjacent radii. h = sin(3*Pi/10) = cos(Pi/5) (radius 1 unit). - _Wolfdieter Lang_, Jan 08 2018
%C A019863 Also the limiting value(L) of "r" which is abscissa of the vertex of the parabola  F(n)*x^2 - F(n+1)*x + F(n + 2)(where F(n)=A000045(n) are the Fibonacci numbers and n>0). - _Burak Muslu_, Feb 24 2021
%H A019863 Stanislav Sykora, <a href="/A019863/b019863.txt">Table of n, a(n) for n = 0..2000</a>
%H A019863 Scott Aaronson, <a href="http://www.scottaaronson.com/blog/?p=1792">The NEW Ten Most Annoying Questions in Quantum Computing</a> (2014).
%H A019863 Boaz Barak, Moritz Hardt, Ishay Haviv, and Anup Rao, <a href="https://citeseerx.ist.psu.edu/pdf/94177f850784d820786d1da5d80a5f18a3b6d41f">Rounding Parallel Repetitions of Unique Games</a> (2008).
%H A019863 Irit Dinur and David Steurer, <a href="https://arxiv.org/abs/1305.1979">Analytical approach to parallel repetition</a>, arXiv:1305.1979 [cs.CC], 2013-2014.
%H A019863 Michael Penn, <a href="https://www.youtube.com/watch?v=EzCBUY6htwk">a golden value of cosine.</a>, YouTube video, 2021.
%H A019863 Wikipedia, <a href="http://en.wikipedia.org/wiki/Exact_trigonometric_constants">Exact trigonometric constants</a>.
%H A019863 Wikipedia, <a href="http://en.wikipedia.org/wiki/Platonic_solid">Platonic solid</a>.
%H A019863 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%F A019863 Equals (1+sqrt(5))/4 = cos(Pi/5) = sin(3*Pi/10). - _R. J. Mathar_, Jun 18 2006
%F A019863 Equals 2F1(4/5,1/5;1/2;3/4) / 2 = A019827 + 1/2. - _R. J. Mathar_, Oct 27 2008
%F A019863 Equals A001622 / 2. - _Stanislav Sykora_, Jan 30 2014
%F A019863 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
%F A019863 Equals 1/A134972. - _R. J. Mathar_, Jan 17 2021
%F A019863 Equals 2*A019836*A019872. - _R. J. Mathar_, Jan 17 2021
%F A019863 Equals (A094214 + 1)/2 or 1/(2*A094214). - _Burak Muslu_, Feb 24 2021
%F A019863 Equals hypergeom([-2/5, -3/5], [6/5], -1) = hypergeom([-1/5, 3/5], [6/5], 1) = hypergeom([1/5, -3/5], [4/5], 1). - _Peter Bala_, Mar 04 2022
%F A019863 Equals Product_{k>=1} (1 - (-1)^k/A001611(k)). - _Amiram Eldar_, Nov 28 2024
%F A019863 Equals 2*A134944 = 3*A134946 = A187426-11/10 = A296182-1. - _Hugo Pfoertner_, Nov 28 2024
%e A019863 0.80901699437494742410229341718281905886015458990288143106772431135263...
%p A019863 convert(sin(3*Pi/10),radical); # _W. Edwin Clark_, May 24, 2023
%p A019863 Digits:=100; evalf((1+sqrt(5))/4); # _Wesley Ivan Hurt_, Mar 27 2014
%t A019863 RealDigits[(1 + Sqrt[5])/4, 10, 111] (* _Robert G. Wilson v_ *)
%t A019863 RealDigits[Sin[54 Degree],10,120][[1]] (* _Harvey P. Dale_, Apr 21 2018 *)
%o A019863 (PARI) (1+sqrt(5))/4 \\ _Charles R Greathouse IV_, Jan 16 2012
%Y A019863 Cf. A001611, A001622, A019827, A134972, A134944, A134946, A187426, A296182.
%Y A019863 Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A239798 (dodecahedron).
%K A019863 nonn,cons,easy
%O A019863 0,1
%A A019863 _N. J. A. Sloane_.

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