A181877 Integer coefficient array for polynomials related to the minimal polynomials of cos(2Pi/n). Rising powers of x.

-2, 2, 2, 2, 1, 2, 0, 2, -1, 2, 4, -1, 2, -1, -4, 4, 8, -2, 0, 4, 1, -6, 0, 8, -1, -2, 4, 1, 6, -12, -32, 16, 32, -3, 0, 4, -1, 6, 24, -32, -80, 32, 64, 1, -4, -4, 8, 1, 8, -16, -8, 16, 2, 0, -16, 0, 16, 1, -8, -40, 80, 240, -192, -448, 128, 256, -1, -6, 0, 8, 1, 10, -40, -160, 240, 672, -448, -1024, 256, 512, 5, 0, -20, 0, 16, 1, -16, 32, 48, -96, -32, 64, -1, 6, 12, -32, -16, 32, -1, -12, 60, 280, -560, -1792, 1792, 4608, -2304, -5120, 1024, 2048, 1, 0, -16, 0, 16, -1, 10, 100, -40, -800, 32, 2240, 0, -2560, 0, 1024, -1, -6, 24, 32, -80, -32, 64, 1, 18, 0, -240, 0, 864, 0, -1152, 0, 512, -7, 0, 56, 0, -112, 0, 64, -1, 14, 112, -448, -2016, 4032, 13440, -15360, -42240, 28160, 67584, -24576, -53248, 8192, 16384, 1, -8, -16, 8, 16

Offset: 1

Author

Wolfdieter Lang, Jan 08 2011

Keywords

• sign
• easy
• tabf

Comments

The sequence of row lengths is d(n)+1, with d(n):=A023022(n), n>=2, and d(1):=1: [2, 2, 2, 2, 3, 2, 4, 3, 4, 3, 6, 3, 7, 4, 5, 5, 9, 4, 10, 5, 7,...].

psi(n,x):=sum(a(n,m)*x^m,m=0..d(n)), with the degree d(n):=A023022(n), n>=2, d(1):=1, equals (2^d(n))*Psi(n,x), with the minimal polynomials Psi(n,x) of cos(2*Pi/n), n>=1. See A181875/A181876 for the rational coefficient array of the monic Psi(n,x).

See A232624 for the (monic integer) minimal polynomials of 2*cos(2*Pi/n), called there MR2(n,x) = psi(n, x/2). - Wolfdieter Lang, Nov 29 2013

Examples

			[-2, 2], [2, 2], [1, 2], [0, 2], [-1, 2, 4], [-1, 2], [-1, -4, 4, 8], [-2, 0, 4], [1, -6, 0, 8], [-1, -2, 4], [1, 6, -12, -32, 16, 32],...
		

References

• I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

Links

• Wolfdieter Lang, A181875/A181876. Minimal polynomials of cos(2Pi/n).
• D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40,3 (1933) 165-6.
• W. Watkins and J. Zeitlin, The Minimal Polynomial of cos(2Pi/n), Am. Math. Monthly 100,5 (1993) 471-4.

Crossrefs

Cf. A023022, A181875, A181876, A183918.

Cf. A232624. - Wolfdieter Lang, Nov 29 2013

Programs

Mathematica

ro[n_] := (cc = CoefficientList[ p = MinimalPolynomial[ Cos[2*(Pi/n)], x], x]; 2^Exponent[p, x]*(cc / Last[cc])); Flatten[ Table[ ro[n], {n, 1, 30}]] (* Jean-François Alcover, Sep 28 2011 *)

Formula

a(n,m) = [x^m]((2^d(n))*Psi(n,x)), with the minimal polynomials Psi(n,x) of cos(2*Pi/n), n>=1. See A181875(n,m)/A181876(n,m) for the rational Psi(n,x) coefficients.