%I #22 Jun 28 2017 20:29:08

%S -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,

%T -12,-32,16,32,-3,0,4,-1,6,24,-32,-80,32,64,1,-4,-4,8,1,8,-16,-8,16,2,

%U 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

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

%C 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,...].

%C 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).

%C 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

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

%H Wolfdieter Lang, <a href="/A181875/a181875.pdf">A181875/A181876. Minimal polynomials of cos(2Pi/n).</a>

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2301023">A Note on Trigonometric Algebraic Numbers</a>, Am. Math. Monthly 40,3 (1933) 165-6.

%H W. Watkins and J. Zeitlin, <a href="http://www.jstor.org/stable/2324301">The Minimal Polynomial of cos(2Pi/n)</a>, Am. Math. Monthly 100,5 (1993) 471-4.

%F 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.

%e [-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],...

%t 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 *)

%Y Cf. A023022, A181875, A181876, A183918.

%Y Cf. A232624. - _Wolfdieter Lang_, Nov 29 2013

%K sign,easy,tabf

%O 1,1

%A _Wolfdieter Lang_, Jan 08 2011