A019916 - OEIS

%I #37 Feb 04 2025 15:21:56

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

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

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

%N Decimal expansion of tan(Pi/10) (angle of 18 degrees).

%C In a regular pentagon inscribed in a unit circle this is the cube of the length of the side divided by 5: (1/5)*(sqrt(3 - phi))^3 with phi from A001622. - _Wolfdieter Lang_, Jan 08 2018

%C Quartic number of denominator 5 and minimal polynomial 5x^4 - 10x^2 + 1. - _Charles R Greathouse IV_, May 13 2019

%H Ivan Panchenko, <a href="/A019916/b019916.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Exact_trigonometric_constants">Exact trigonometric constants</a>

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

%F Equals A019827/A019881 = 1/A019970 = 1/sqrt(5+2*sqrt(5)). - _R. J. Mathar_, Jul 26 2010

%F Equals tan((phi - 1)/sqrt(2 + phi)) = (1/5)*(sqrt(3 - phi))^3 = (3 - phi)*sqrt(3 - phi)/5 = sqrt(7 - 4*phi)/(2*phi - 1), with phi from A001622. - _Wolfdieter Lang_, Jan 08 2018

%F Equals Product_{k>=0} ((5*k + 1)/(5*k + 4))^(-1)^(k) = Product_{k>=0} A090771(k)/A090773(k). - _Antonio Graciá Llorente_, Mar 24 2024

%e 0.3249196962329063261558714122151344649549034715214751003078047191...

%t RealDigits[Tan[18 Degree],10,120][[1]] (* _Harvey P. Dale_, Mar 07 2012 *)

%o (PARI) tan(Pi/10) \\ _Michel Marcus_, Jan 08 2018

%o (PARI) polrootsreal(5*x^4-10*x^2+1)[3] \\ _Charles R Greathouse IV_, Feb 04 2025

%Y Cf. A001622, A019827 (sin(Pi/10)), A019881 (cos(Pi/10)).

%K nonn,cons

%O 0,1

%A _N. J. A. Sloane_