A300074 - OEIS

%I #39 Nov 23 2024 04:30:43

%S 8,5,0,6,5,0,8,0,8,3,5,2,0,3,9,9,3,2,1,8,1,5,4,0,4,9,7,0,6,3,0,1,1,0,

%T 7,2,2,4,0,4,0,1,4,0,3,7,6,4,8,1,6,8,8,1,8,3,6,7,4,0,2,4,2,3,7,7,8,8,

%U 4,0,4,7,3,6,3,9,5,8,9,6,6,6,9,4,3,2,0,3,6,4,2,7,8,5,1,7,6

%N Decimal expansion of 1/(2*sin(Pi/5)) = A121570/2.

%C This is the reciprocal of A182007, and one half of A121570.

%C This is the ratio of the radius r of the circumscribing circle of a regular pentagon and its side length s: r/s = 1/(2*sin(Pi/5)).

%C A quartic number of denominator 5 and minimal polynomial 5x^4 - 5x^2 + 1. - _Charles R Greathouse IV_, Mar 04 2018

%C Appears at Schur decomposition of A=[1 2; 2 3]. - _Donghwi Park_, Jun 20 2018

%H Muniru A Asiru, <a href="/A300074/b300074.txt">Table of n, a(n) for n = 0..2000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Schur_decomposition">Schur decomposition</a>.

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.

%F r/s = 1/A182007 = A121570/2 = (2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = (1 + sqrt(5))/2 = A001622.

%F From _Amiram Eldar_, Feb 08 2022: (Start)

%F Equals cos(arccot(phi)) = cos(arctan(1/phi)) = cos(A195693).

%F Equals sin(arctan(phi)) = sin(arccot(1/phi)) = sin(A195723). (End)

%F Equals Product_{k>=1} (1 + (-1)^k/A090773(k)). - _Amiram Eldar_, Nov 23 2024

%e r/s = 0.850650808352039932181540497063011072240401403764816881836740242377...

%e 2*r/s = A121570.

%t RealDigits[1/(2 Sin[Pi/5]), 10, 111][[1]] (* _Robert G. Wilson v_, Jul 15 2018 *)

%o (PARI) 1/(2*sin(Pi/5)) \\ _Charles R Greathouse IV_, Mar 04 2018

%o (PARI) sqrt((5+sqrt(5))/10) \\ _Charles R Greathouse IV_, Mar 04 2018

%Y Cf. A001622, A090773, A121570, A182007, A195693, A195723.

%K nonn,cons,easy

%O 0,1

%A _Wolfdieter Lang_, Mar 01 2018