cp's OEIS Frontend

A136571 Irregular triangle of coefficients of the minimal polynomial of 2*cos(2*Pi/n) in decreasing powers.

%I A136571 #11 Feb 16 2025 08:33:07
%S A136571 1,-2,1,2,1,1,1,0,1,1,-1,1,-1,1,1,-2,-1,1,0,-2,1,0,-3,1,1,-1,-1,1,1,
%T A136571 -4,-3,3,1,1,0,-3,1,1,-5,-4,6,3,-1,1,-1,-2,1,1,-1,-4,4,1,1,0,-4,0,2,1,
%U A136571 1,-7,-6,15,10,-10,-4,1,1,0,-3,-1,1,1,-8,-7,21
%N A136571 Irregular triangle of coefficients of the minimal polynomial of 2*cos(2*Pi/n) in decreasing powers.
%C A136571 The degree of the n-th polynomial is A023022(n), the half-totient function for n>2. These polynomials are integral, monic and irreducible over the integers. Hence 2*cos(2*Pi/n) is an algebraic integer. When n is prime, the n-th row is the same as the n-th row of A066170. Carlitz and Thomas give an algorithm for computing these polynomials.
%H A136571 T. D. Noe, <a href="/A136571/b136571.txt">Rows n=1..100 of triangle, flattened</a>
%H A136571 Scott Beslin and Valerio de Angelis, <a href="http://www.jstor.org/stable/3219105">The minimal polynomials of sin(2 pi/p) and cos(2 pi/n)</a>, Math. Mag., 77 (2004), 146-149.
%H A136571 L. Carlitz and J. M. Thomas, <a href="http://www.jstor.org/stable/2310783">Rational tabulated values of trigonometric functions</a>, Amer. Math. Monthly, 69 (1962), 789-793.
%H A136571 G. P. Dresden, <a href="http://www.jstor.org/stable/4145075">On the middle coefficient of a cyclotomic polynomial</a>, Amer. Math. Monthly, 111 (No. 6, 2004), 531-533.
%H A136571 D. H. Lehmer, <a href="http://www.jstor.org/stable/2301023">A note on trigonometric algebraic numbers</a>, Amer. Math. Monthly, 40 (1933), 165-166.
%H A136571 William Watkins and Joel Zeitlin, <a href="http://www.jstor.org/stable/2324301">The minimal polynomial of cos(2 pi/n)</a>, Amer. Math. Monthly, 100 (1993), 471-474.
%H A136571 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrigonometryAngles.html">Trigonometry Angles</a>
%e A136571 x-2, x+2, x+1, x, x^2+x-1, x-1, x^3+x^2-2x-1, x^2-2, x^3-3x+1, x^2-x-1
%t A136571 Flatten[Table[Reverse[CoefficientList[MinimalPolynomial[2Cos[2Pi/n],x],x]], {n,25}]]
%K A136571 nice,sign,tabf
%O A136571 1,2
%A A136571 _T. D. Noe_, Jan 07 2008