OFFSET
1,3
COMMENTS
Also degree of minimal polynomial of function F(n)=(gamma(1/n)*gamma((n-1)/n))/Pi over the rationals. Roots of minimal polynomials of F(n) belonging to algebraic extension of sin(n/Pi) and vice versa (e.g. gamma(1/11)*gamma(10/11)/Pi = 20*sin(Pi/11) - 112*sin(Pi/11)^3 + 256*sin(Pi/11)^5 - 256*sin(Pi/11)^7 + (1024*sin(Pi/11)^9)/11). - Artur Jasinski, Oct 17 2011
The algebraic numbers sin(Pi/(2*l)) are given in A228783 in the power basis of the number field Q(2*cos(Pi/(2*l))) if n is even and of Q(2*cos(Pi/l)) if l is odd. In A228785, sin(Pi/(2*l+1)) is given in the power basis of Q(2*cos(Pi/(2*(2*l+1)))) (only odd powers appear). The minimal polynomials for 2*sin(Pi/n), n>=1, are given in A228786. - Wolfdieter Lang, Oct 10 2013
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..175
Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.
Eric Weisstein's World of Mathematics, Trigonometry Angles
FORMULA
a(1)=1, a(2)=1, a(n)=phi(n)/(1, 1, 2, 1 for n=0, 1, 2, 3 mod 4) for n>2, where phi is Euler's totient, A000010
a(n) = A093819(2*n), n >= 1.- Wolfdieter Lang, Oct 29 2019
MATHEMATICA
a[n_] := If[n==2, 1, EulerPhi[n]/{1, 1, 2, 1}[[Mod[n, 4]+1]]]; Table[a[n], {n, 80}]
a[n_] := Exponent[ MinimalPolynomial[Sin[Pi/n]][x], x]; Array[a, 75] (* Robert G. Wilson v, Jul 28 2014 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Shawn Cokus (Cokus(AT)math.washington.edu)
STATUS
approved