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%I #98 Dec 01 2021 20:10:39
%S -2,1,2,1,1,1,0,1,-1,1,1,-1,1,-1,-2,1,1,-2,0,1,1,-3,0,1,-1,-1,1,1,3,
%T -3,-4,1,1,-3,0,1,-1,3,6,-4,-5,1,1,1,-2,-1,1,1,4,-4,-1,1,2,0,-4,0,1,1,
%U -4,-10,10,15,-6,-7,1,1,-1,-3,0,1,1,5,-10,-20,15,21,-7,-8,1,1,5,0,-5,0,1,1,-8,8,6,-6,-1,1,-1,3,3,-4,-1,1
%N Coefficient array for the minimal polynomials of 2*cos(2*Pi/n) for n >= 1.
%C The length of row n is deg(n) + 1, n >= 1, with the degree deg(1) = deg(2) = 1, and deg(n) = phi(n)/2 = A023022(n) for n >= 3. That is: 2, 2, 2, 2, 3, 2, 4, 3, 4, 3, 6, 3, 7, 4, 5, 5, 9, 4, 10, 5, ...
%C 2*cos(2*Pi/n) = R(2, rho(n)) = -2 + rho(n)^2, with rho(n) = 2*cos(Pi/n) and the monic Chebyshev T-polynomials R(n, x), n>=1, with coefficient table A127672. For even n 2*cos(2*Pi/n) becomes rho(n/2). Therefore, 2*cos(2*Pi/n) is an integer in the algebraic number field Q(rho(n/2)) or Q(rho(n)) if n is even or odd, respectively. The degree deg(n) of the minimal polynomials, call them MPR2(n, x), is delta(n/2) or delta(n) for even or odd n, respectively, with delta(n) = A055034(n). This becomes deg(n) as given above.
%C These minimal polynomials are C(n/2, x) if n is even, with C(k, x) the minimal polynomials of rho(k) given in A187360.
%C For odd n the known zeros of C(n, x) are rho(n) and its conjugates, call them rho(n;j), j=1, 2, ..., delta(n), with rho(n;1) = rho(n). These conjugates can be written in the power basis of Q(rho(2*l+1)), l >= 1. See the link to the Q(2cos(Pi/n)) paper in A187360, and there Table 4. Then the (monic) minimal polynomial MPR2(2*l+1, x) = Product_{j=1..delta(2*l+1)} (x - (-2 + rho(2*l+1;j)^2)), l >= 0. After expansion all powers of rho(2*l+1) not smaller than delta(2*l+1) are reduced with the help of C(2*l+1,rho(2*l+1)) = 0, leading automatically to integer coefficients (without using the trigonometric version of rho(2*l+1)).
%C Compare the present minimal polynomials with the (non-monic) minimal polynomials of cos(2*Pi/n) given in an _Artur Jasinski_ comment from Oct 28 2008 on A023022.
%C The present monic integer minimal polynomials of 2*cos(2*Pi/n), called MPR2(n, x), are related to the non-monic integer minimal polynomials of 2*cos(2*Pi/n) of A181877, called there psi(n, x) by MPR2(n, x) = psi(n, x/2). See Table 5 of the Wolfdieter Lang link given there. - _Wolfdieter Lang_, Nov 29 2013
%C The present minimal polynomials MPR2(n, x) are C(n/2, x) if n is even (see above) and (-1)^degC(n)*C(n, -x) if n is odd, with the C polynomials from A187360 of degree degC(n) = A055034(n). Note that degC(2*k+1) = deg(2*k+1) = A023022(2*k+1), k >= 0. - _Wolfdieter Lang_, Apr 12 2018
%C Let {U(n, x)} be defined as: U(0, x) = 0, U(1, x) = 1, U(n, x) = x*U(n-1, x) - U(n-2, x) for n >= 2, then U(n, x) = Product_{k|2n, k>=3} MPR2(k, x) for n > 0, because U(n, x) = Product_{m=1..n-1} (x - 2*cos(Pi*m/n)) for n > 0. - _Jianing Song_, Jul 08 2019
%C Conjecture: For odd n > 1, the term of the highest degree of (MPR2(2n, x) - MPR2(n, x))/2 is (-1)^omega(n) * x^(phi(n)/2-n/rad(n)) = A076479(n) * x^(A023022(n)-A003557(n)). For example, for n = 15, (MPR2(30, x) - MPR2(15, x))/2 = x^3 - 4x; for n = 105, (MPR2(210, x) - MPR2(105, x))/2 = -x^23 + ...; for n = 225, (MPR2(450, x) - MPR2(225, x))/2 = x^45 + ... If this is true, then for odd n > 1, a(n,A023022(n)-k) = a(2n,A023022(n)-k) = 0 for k = 1, 3, ..., A003557(n)-2; a(n,A023022(n)-A003557(n)) = -A076479(n) and a(2n,A023022(n)-A003557(n)) = A076479(n). - _Jianing Song_, Jul 11 2019
%C Conjecture: Let MPR2(n, x) equal the odd indexed (n) monic polynomial. If the number of roots with negative signs is even, then n is a term in A014659. Example: n = 7 for x^3 + x^2 - 2x - 1, having two negative roots, (-445041..., and -1.801937...). Two is even so the integer 7 is in A014659. n = 9 for the polynomial x^3 - 3x + 1, with one negative root, (-1.87938). The term 9 is in A014657. - _Gary W. Adamson_, Oct 20 2021
%C From _Gary W. Adamson_, Nov 30 2021 (Start)
%C Given the first (phi(n))/2 terms for odd n, the number of even terms in the set is equal to the number of positive roots in MPR2(n, x). The number of odd terms is equal to the number of negative roots in MPR2(n, x). For n = 11, (phi(11))/2 = 5, and the set is (1, 2, 3, 4, 5); having two even and three odd terms.
%C Given MPR2(11, x) = x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1, there are two roots with positive signs: 1.682508..., and .830830...; and three roots with negative signs: -1.918985..., -1.309921..., and -.284629....Using the Descartes' rule for signs, MPR2(11, x) has coefficients signed (+ + - - + +); having two sign changes indicating two positive roots. With all real roots there are three (= 5 - 2) roots signed negative. (End)
%H Michael De Vlieger, <a href="/A232624/b232624.txt">Table of n, a(n) for n = 1..14000</a> (rows 1 <= n <= 300, flattened)
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/2008.04300">On the Equivalence of Three Complete Cyclic Systems of Integers</a>, arXiv:2008.04300 [math.NT], 2020.
%F a(n,m) = [x^m] MPR2(n, x), n >= 1, m=0, 1, ..., deg(n), with MPR2(n, x) the (monic) minimal polynomials of 2*cos(2*Pi/n), explained in a comment above. The degree is deg(1) = deg(2) = 1, deg(n) = phi(n)/2 = A023022(n), n >= 3 (phi is the Euler totient function A000010).
%F From _Jianing Song_, Jul 09 2019: (Start)
%F MPR2(n, x) = Product_{0<=m<=n/2, gcd(m, n)=1} (x - 2*cos(2*Pi*m/n)).
%F If 4 divides n, then MPR2(n, x) = Product_{k|(n/2)} U((n/2)/k, x)^mu(k), where U(n, x) is the polynomial defined in comment and mu = A008683. For odd n, MPR2(n, x)*MPR2(2n, x) = Product_{k|n} U(n/k, x)^mu(k).
%F If 4 divides n and n > 4, then a(n,2k+1) = 0, that is, MPR2(n, x) contains even powers of x only.
%F For odd n > 1, a(2n,k) = a(n,k)*(-1)^(A023022(n)-k). (End)
%e The table a(n,m) begins:
%e n\m 0 1 2 3 4 5 6 7 8 9 ...
%e 1: -2 1
%e 2: 2 1
%e 3: 1 1
%e 4: 0 1
%e 5: -1 1 1
%e 6: -1 1
%e 7: -1 -2 1 1
%e 8: -2 0 1
%e 9: 1 -3 0 1
%e 10: -1 -1 1
%e 11: 1 3 -3 -4 1 1
%e 12: -3 0 1
%e 13: -1 3 6 -4 -5 1 1
%e 14: 1 -2 -1 1
%e 15: 1 4 -4 -1 1
%e 16: 2 0 -4 0 1
%e 17: 1 -4 -10 10 15 -6 -7 1 1
%e 18: -1 -3 0 1
%e 19: 1 5 -10 -20 15 21 -7 -8 1 1
%e 20: 5 0 -5 0 1
%e ...
%e MPR2(14, x) = C(7, x) = 1 - 2*x - x^2 + x^3.
%e MPR2(7, x) = (x - (-2 + z^2))*(x - (-2 + (-1 - z + z^2)^2))*
%e (x - (-2 + (2 - z^2)^2)), with z = rho(7). Expanded and reduced with C(7, z) = 0 this becomes finally MPR2(7, x) = -1 - 2*x + x^2 + x^3.
%e MPR2(7, x) = -C(7, -x). - _Wolfdieter Lang_, Apr 12 2018
%t ro[n_] := (MPR2 = CoefficientList[p = MinimalPolynomial[2*Cos[2*(Pi/n)], x], x]; MPR2); Flatten[Table[ro[n], {n, 30}]] (* _Jianing Song_, Jul 09 2019 *)
%Y Cf. A023022 (degree), A055034, A187360 (C polynomials).
%Y Cf. A181877, A181875/A181876. - _Wolfdieter Lang_, Nov 29 2013
%Y Cf. A014657/A014659.
%Y Cf. A065941.
%Y Cf. A003558.
%K sign,tabf,easy
%O 1,1
%A _Wolfdieter Lang_, Nov 28 2013
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