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%I A232630 #9 Dec 18 2013 19:56:45 %S A232630 0,1,0,1,-3,0,1,0,1,5,0,-5,0,1,-3,0,1,-7,0,14,0,-7,0,1,-2,1,-3,0,9,0, %T A232630 -6,0,1,5,0,-5,0,1,-11,0,55,0,-77,0,44,0,-11,0,1,-3,0,1,13,0,-91,0, %U A232630 182,0,-156,0,65,0,-13,0,1,-7,0,14,0,-7,0,1,1,0,-8,0,14,0,-7,0,1,-2,0,1,17,0,-204,0,714,0,-1122,0,935,0,-442,0,119,0,-17,0,1 %N A232630 Coefficient table for the minimal polynomials of 2*sin(4*Pi/n). Rising powers of x. %C A232630 The length of row n is A232626(n) + 1, that is 2, 2, 3, 2, 5, 3, 7, 2, 7, 5, 11, 3, 13, 7, 9, 3, 17, 7, 19, 5,... %C A232630 In a regular n-gon, n>=2, inscribed in a circle of radius R (in some length units), 2*sin(4*Pi/n) = (S(n)/R)*(D(1,n)/S(n)) = D(1,n)/R, with the side length S(n) and the length of the first (smallest) diagonal D(1,n). For n=2 there is no such diagonal, and one can put D(1,2) = 0. Obviously, D(1,2*m) = S(m), m >= 2. %C A232630 For the power basis representation of 2*sin(4*Pi/n) in the algebraic number field Q(rho(q(2,n))), with q(2,n)) = A232625(n) and rho(m) := 2*cos(Pi/m), see A232629. Call the row polynomials of A232629 PB2(n,x) (power basis polynomial for the case k=2 in 2*sin(2*Pi*k/n)). %C A232630 The minimal polynomial of 2*sin(4*Pi/n), call it MP2(n, x), is obtained from the conjugates rho(q(2,n),j), j= 1, ... , delta(q(2,n)) = A232626(n), which are the zeros of C(q(2,n), x), the minimal polynomial of rho(q(2,n)) = rho(q(2,n),1) (for C see A187360). MP2(n, x) = product(x - PB2(n, rho(q(2,n),j)), j=1..A232626(n)) (mod C(q(2,n), rho(q(2,n)))). %F A232630 a(n,m) = [x^m] MP2(n, x), n>=1, m = 0, 1, ..., A232626(n), with the minimal polynomials of 2*sin(4*Pi/n), computed like explained above in a comment. %F A232630 a(2*l,m) = A231188(l,m), m = 0, 1, ..., A093819(l), l >= 1. %e A232630 The table a(n,m) begins: %e A232630 -------------------------------------------------------------------------------------- %e A232630 n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ... %e A232630 1: 0 1 %e A232630 2: 0 1 %e A232630 3: -3 0 1 %e A232630 4: 0 1 %e A232630 5: 5 0 -5 0 1 %e A232630 6: -3 0 1 %e A232630 7: -7 0 14 0 -7 0 1 %e A232630 8: -2 1 %e A232630 9: -3 0 9 0 -6 0 1 %e A232630 10: 5 0 -5 0 1 %e A232630 11: -11 0 55 0 -77 0 44 0 -11 0 1 %e A232630 12: -3 0 1 %e A232630 13: 13 0 -91 0 182 0 -156 0 65 0 -13 0 1 %e A232630 14: -7 0 14 0 -7 0 1 %e A232630 15: 1 0 -8 0 14 0 -7 0 1 %e A232630 16: -2 0 1 %e A232630 17: 17 0 -204 0 714 0 -1122 0 935 0 -442 0 119 0 -17 0 1 %e A232630 18: -3 0 9 0 -6 0 1 %e A232630 19: -19 0 285 0 -1254 0 2508 0 -2717 0 1729 0 -665 0 152 0 -19 0 1 %e A232630 20: 5 0 -5 0 1 %e A232630 ... %e A232630 n=1: 2*sin(4*Pi/1) = 0 is rational, therefore MP2(1, x) = x, with coefficients 0, 1, and degree A232626(1) = 1. PB2(1, rho(1,1)) = PB2(1, rho(1)) = 0. %e A232630 n=3: A232626(2) = 2. PB2(2, x) = -x, C(6, x) = x^2 - 3, with zeros rho(6) and R(5, rho(6)) (for R see A127672), hence rho(6,1) = rho(6) and rho(6,2) = R(5, rho(6))= 5*rho(6) - 5*rho(6)^3 + 1*rho(6)^5, MP2(3, x) = (x - (-rho(6)))*(x - (- R(5, rho(6))) reduced with rho(6)^2 = 3 leading to MP2(3, x) = -3 + x^2, yielding row n=3: -3 0 1. %e A232630 n=8: this row -2, 1 coincides with row n=4 of A231188. %e A232630 n=17: coincides with WolframAlpha's MinimalPolynomial[2*sin(4*Pi/17),x] = 17-204 x^2+714 x^4-1122 x^6+935 x^8-442 x^10+119 x^12-17 x^14+x^16. %Y A232630 Cf. A231188 (k=1 case), A187360 (C), A127672(R), A232626 (degree), A232629 (PB2). %K A232630 sign,tabf,easy %O A232630 1,5 %A A232630 _Wolfdieter Lang_, Dec 17 2013