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Numerators of coefficient array for minimal polynomials of sin(2*Pi/n). Rising powers of x.
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%I #42 Apr 29 2021 12:03:46

%S 0,1,0,1,-3,0,1,-1,1,5,0,-5,0,1,-3,0,1,-7,0,7,0,-7,0,1,-1,0,1,-3,0,9,

%T 0,-3,0,1,5,0,-5,0,1,-11,0,55,0,-77,0,11,0,-11,0,1,-1,1,13,0,-91,0,91,

%U 0,-39,0,65,0,-13,0,1,-7,0,7,0,-7,0,1,1,0,-1,0,7,0,-7,0,1,1,0,-1,0,1,17,0,-51,0,357,0,-561,0,935,0,-221,0,119,0,-17,0,1,-3,0,9,0,-3,0,1,-19,0,285,0,-627,0,627,0,-2717,0,1729,0,-665,0,19,0,-19,0,1,-1,1,1,1,0,-1,0,15,0,-39,0,11,0,-11,0,1,-11,0,55,0,-77,0,11,0,-11,0,1

%N Numerators of coefficient array for minimal polynomials of sin(2*Pi/n). Rising powers of x.

%C The corresponding denominator array is given in A181873(n,m).

%C The sequence of row lengths of this array is A093819(n)+1: [2, 2, 3, 2, 5, 3, 7, 3, 7, 5, 11, ...].

%C The minimal polynomial of the algebraic number sin(2*Pi/n), n >= 1, is here called Pi(n,x) := Sum_{m=0..d(n)} r(n,m)*x^m with the degree sequence d(n):=A093819(n), and the rationals r(n):=a(n,m)/b(n,m) with b(n,m):=A181873(n,m).

%C See the Niven reference, p. 28, for the definition of 'minimal polynomial of an algebraic number'.

%C Minimal polynomials are irreducible.

%C The minimal polynomials of sin(2*Pi/n) are treated, e.g., in the Lehmer, Niven and Watkins-Zeitlin references.

%C The minimal polynomials Pi(n,x) of sin(2*Pi/n) are found from Psi(c(n),x), where Psi(m,x) is the minimal polynomial of cos(2*Pi/m), and

%C c(n):= denominator(|(4-n)/(4*n)|) = A178182(n).

%C For the regular n-gon inscribed in the unit circle the area is n*sin(2*Pi/n). See the remark by _Jack W Grahl_ under A093819.

%C S. Beslin and V. de Angelis (see the reference) give an explicit formula for the (integer) minimal polynomial of sin(2*Pi/p), called S_p(x), and cos(2*Pi/p), called C_p(x),for odd prime p, p=2k+1, with the results:

%C S_p(x) = Sum_{l=0..k} ((-1)^l)*binomial(p,2*l+1)*(1-x^2)^(k-l)*x^(2*l), and C_p(x) = S_p(sqrt((1-x)/2)), where S_p(x), with leading term ((-2)^k))*x^(p-1), checks with((-2)^k)*Pi(p,x). - _Wolfdieter Lang_, Feb 28 2011

%C The zeros of Pi(n, x) result from those of the minimal polynomial Psi(n, x) of cos(2*Pi/n), and they are cos(2*Pi*k/n), for k = 0, ..., floor(c(n)/2), with c(n) = A178182(n), and restriction gcd(k, c(n)) = 1, for n >= 1. There are d(n) = A093819(n) such zeros. - _Wolfdieter Lang_, Oct 30 2019

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

%H S. Beslin and V. de Angelis, <a href="https://www.jstor.org/stable/3219105">The minimal Polynomials of sin(2Pi/p) and cos(2Pi/p)</a>, Mathematics Mag. 77.2 (2004) 146-9.

%H Wolfdieter Lang, <a href="/A181872/a181872_1.pdf">Minimal polynomials for sin(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) = numerator([x^m]Pi(n,x)), n>=1, m=0..A093819(n). For Pi(n,x) see the comments.

%F The minimal polynomial Pi(n,x) = Product_{k=0..floor(c(n)/2), gcd(k, c(n)) = 1}, x - cos(2*Pi*k/c(n)), for n >= 1. - _Wolfdieter Lang_, Oct 30 2019

%e Triangle begins:

%e [0, 1],

%e [0, 1],

%e [-3, 0, 1],

%e [-1, 1],

%e [5, 0, -5, 0, 1],

%e [-3, 0, 1],

%e [-7, 0, 7, 0, -7, 0, 1],

%e [-1, 0, 1],

%e [-3, 0, 9, 0, -3, 0, 1],

%e [5, 0, -5, 0, 1],

%e ...

%e The rational coefficients r(n,m) start like:

%e [0, 1],

%e [0, 1],

%e [-3/4, 0, 1],

%e [-1, 1],

%e [5/16, 0, -5/4, 0, 1],

%e [-3/4, 0, 1],

%e [-7/64, 0, 7/8, 0, -7/4, 0, 1],

%e [-1/2, 0, 1],

%e [-3/64, 0, 9/16, 0, -3/2, 0, 1],

%e ...

%e Pi(6,n) = Psi(c(6),x) = Psi(12,x) = x^2-3/4.

%t p[n_, x_] := MinimalPolynomial[ Sin[2 Pi/n], x]; Flatten[ Numerator[ Table[ coes = CoefficientList[ p[n, x], x]; coes / Last[coes], {n, 1, 22}]]] (* _Jean-François Alcover_, Nov 07 2011 *)

%Y Cf. A181875, A181876 (minimal polynomials of cos(2*Pi/n)).

%K sign,easy,frac,tabf

%O 1,5

%A _Wolfdieter Lang_, Jan 13 2011