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

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, -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, 1, -7, -6, 15, 10, -10, -4, 1, 1, 0, -3, -1, 1, 1, -8, -7, 21

Offset: 1

Author

T. D. Noe, Jan 07 2008

Keywords

• nice
• sign
• tabf

Comments

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.

Examples

			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
		

Links

• T. D. Noe, Rows n=1..100 of triangle, flattened
• Scott Beslin and Valerio de Angelis, The minimal polynomials of sin(2 pi/p) and cos(2 pi/n), Math. Mag., 77 (2004), 146-149.
• L. Carlitz and J. M. Thomas, Rational tabulated values of trigonometric functions, Amer. Math. Monthly, 69 (1962), 789-793.
• G. P. Dresden, On the middle coefficient of a cyclotomic polynomial, Amer. Math. Monthly, 111 (No. 6, 2004), 531-533.
• D. H. Lehmer, A note on trigonometric algebraic numbers, Amer. Math. Monthly, 40 (1933), 165-166.
• William Watkins and Joel Zeitlin, The minimal polynomial of cos(2 pi/n), Amer. Math. Monthly, 100 (1993), 471-474.
• Eric Weisstein's World of Mathematics, Trigonometry Angles

Programs

Mathematica

Flatten[Table[Reverse[CoefficientList[MinimalPolynomial[2Cos[2Pi/n],x],x]], {n,25}]]