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%I A181875 #41 Oct 02 2023 13:48:24
%S A181875 -1,1,1,1,1,1,0,1,-1,1,1,-1,1,-1,-1,1,1,-1,0,1,1,-3,0,1,-1,-1,1,1,3,
%T A181875 -3,-1,1,1,-3,0,1,-1,3,3,-1,-5,1,1,1,-1,-1,1,1,1,-1,-1,1,1,0,-1,0,1,1,
%U A181875 -1,-5,5,15,-3,-7,1,1,-1,-3,0,1,1,5,-5,-5,15,21,-7,-2,1,1,5,0,-5,0,1,1,-1,1,3,-3,-1,1,-1,3,3,-1,-1,1,-1,-3,15,35,-35,-7,7,9,-9,-5,1,1,1,0,-1,0,1,-1,5,25,-5,-25,1,35,0,-5,0,1,-1,-3,3,1,-5,-1,1,1,9,0,-15,0,27,0,-9,0,1,-7,0,7,0,-7,0,1,-1,7,7,-7,-63,63,105,-15,-165,55,33,-3,-13,1,1,1,-1,-1,1,1
%N A181875 Numerator of coefficient array of minimal polynomials of cos(2Pi/n). Rising powers in x.
%C A181875 The corresponding denominator array is A181876(n,m).
%C A181875 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 A181875 Psi(n,x):=sum((a(n,m)/b(n,m))*x^m,m=0..d(n)), with the degree d(n):=A023022(n), n>=2, d(1):=1, and b(n,m):=A181876(n,m), is the minimal polynomial of cos(2*Pi/n), n>=1. For the definition of `minimal polynomial of an algebraic number' see, e.g., the I. Niven reference, p. 28 (monic, minimal degree rational polynomial with the algebraic number as one of its roots).
%C A181875 All the roots of the minimal polynomial Psi(n,x), are cos(2*Pi*k/n) for k from {0,1,...,floor(n/2)} and gcd(k,n)=1 (relatively prime). The degree d(n) (see above) of Psi(n,x), hence of the algebraic number cos(2*Pi/n), is 1 for n=1 and 2, and phi(n)/2 for n>2, with Euler's totient function phi(n)=A000010(n). See the D. H. Lehmer reference, and the I. Niven reference, Theorem 3.9, p. 37. This is the Lemma on p. 473 of the Watkins and Zeitlin reference (including the n=1 and n=2 cases).
%C A181875 A recurrence for Psi(n,x) is found in the Watkins and Zeitlin reference.
%C A181875 For the solution of the Watkins and Zeitlin recurrence see the W.Lang link under A007955, eqs. (1) and (3), and the theorem with proposition 1. W. Lang, Feb 26 2011.
%C A181875 The polynomials Psi(n,x), n=1..30, have been given in a comment on A023022 by A. Jasinski. See also the W. Lang link.
%C A181875 For powers of each prime number p one finds the following results for m=1,2,...:
%C A181875 1. p odd prime,p=2*k+1:(2^(k*p^(m-1)))*Psi(p^m,x) = 2*sum(T(l*p^(m-1),x),l=1..k) + 1, with Chebyshev's T-polynomials.
%C A181875 2. p=2, m=1: Psi(2,x) = x+1 = T(1,x) + 1.
%C A181875 For m=2,3,...:(2^(m-2))*Psi(2^m,x) = 2*T(2^(m-2),x).
%C A181875 For some odd p the case m=1 has been observed in an e-mail by G. Detlefs to W. Lang. Feb 26 2011.
%C A181875 For the proofs see the W. Lang link, note added.
%C A181875 D. Surowski and P. McCombs (see the reference) give in their theorem 3.1. an explicit formula for the (non-monic) minimal polynomial of 2*cos(2*Pi/p) for odd prime p, p=2*k+1, called Theta_p(x). Their formula checks with Theta_p(x)=(2^k)*Psi(p,x/2) (if the misprint sigma_{2k+1} is corrected to sigma_{2k-1}).
%C A181875 W. Lang, Feb 26 2011.
%C A181875 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 A181875 S_p(x) = sum(((-1)^l)*binomial(p,2*l+1)*(1-x^2)^(k-l) *x^(2*l),l=0..k), and C_p(x) = S_p(sqrt((1-x)/2)).
%C A181875 C_p(x) checks with (2^k)*Psi(p,x) from the above formula for powers of p, with m=1. W. Lang, Feb 26 2011.
%D A181875 I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.
%H A181875 S. Beslin and V. de Angelis, The minimal Polynomials of sin(2pi/p) and cos(2pi/p), Mathematics Mag. 77.2 (2004) 146-9.
%H A181875 Wolfdieter Lang, A181875/A181876. Minimal polynomials of cos(2Pi/n).
%H A181875 Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv preprint arXiv:1210.1018 [math.GR], 2012-2017. - From _N. J. A. Sloane_, Dec 30 2012
%H A181875 D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40,3 (1933) 165-6.
%H A181875 D. Surowski and P. McCombs, Homogeneous Polynomials and the Minimal Polynomials of cos(2pi/n), Missouri J. of Math. Sciences, 15,1 (2003) 4-14.
%H A181875 W. Watkins and J. Zeitlin, The Minimal Polynomial of cos(2Pi/n), Am. Math. Monthly 100,5 (1993) 471-4.
%F A181875 a(n,m) = numerator([x]^m Psi(n,x)), n>=1, m=0,1,..,d(n), with d(n):=A023022(n) and d(1):=1, where Psi(n,x) has been defined in the comment above and is given by Psi(n,x)= product(x-cos(2*Pi*k/n)),k=0..floor(n/2)and gcd(k,n)=1), n>=1.
%e A181875 Rows begin:
%e A181875 [-1, 1],
%e A181875 [1, 1],
%e A181875 [1, 1],
%e A181875 [0, 1],
%e A181875 [-1, 1, 1],
%e A181875 [-1, 1],
%e A181875 [-1, -1, 1, 1],
%e A181875 [-1, 0, 1],
%e A181875 [1, -3, 0, 1],
%e A181875 [-1, -1, 1],
%e A181875 ...
%e A181875 Array of rationals a(n,m)/A181876(n,m):
%e A181875 [-1, 1],
%e A181875 [1, 1],
%e A181875 [1/2, 1],
%e A181875 [0, 1],
%e A181875 [-1/4, 1/2, 1],
%e A181875 [-1/2, 1],
%e A181875 [-1/8, -1/2, 1/2, 1],
%e A181875 [-1/2, 0, 1],
%e A181875 [1/8, -3/4, 0, 1],
%e A181875 [-1/4, -1/2, 1],
%e A181875 ...
%e A181875 Psi(5,x) has the zeros cos(2*Pi/5)=(phi-1)/2 and cos(4*Pi/5)=-phi/2 with phi:=(1+sqrt(5))/2 (golden section).
%t A181875 ro[n_] := Numerator[ cc = CoefficientList[ MinimalPolynomial[ Cos[2*Pi/n], x], x] ; cc / Last[cc]]; Flatten[ Table[ ro[n], {n, 1, 30}]] (* _Jean-François Alcover_, Sep 27 2011 *)
%Y A181875 Cf. A181876, A181877, A023022, A183918.
%K A181875 sign,easy,tabf
%O A181875 1,22
%A A181875 _Wolfdieter Lang_, Jan 08 2011
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