A187360 Coefficient array for minimal polynomials of 2*cos(Pi/n) (rising powers of x).

2, 1, 0, 1, -1, 1, -2, 0, 1, -1, -1, 1, -3, 0, 1, 1, -2, -1, 1, 2, 0, -4, 0, 1, -1, -3, 0, 1, 5, 0, -5, 0, 1, -1, 3, 3, -4, -1, 1, 1, 0, -4, 0, 1, -1, -3, 6, 4, -5, -1, 1, -7, 0, 14, 0, -7, 0, 1, 1, -4, -4, 1, 1, 2, 0, -16, 0, 20, 0, -8, 0, 1, 1, 4, -10, -10, 15, 6, -7, -1, 1

Offset: 1

Author

Wolfdieter Lang, Jul 14 2011

Keywords

• sign
• easy
• tabf

Comments

The degree delta(n) of the monic integer row polynomial, call it C(n,x), is A055034(n).

This minimal polynomial of the algebraic number 2*cos(Pi/n), n>=1, is given by

C(n,x) = sum(a(n,m)*x^m,m=0..A055034(n)) = (2^delta(n))*Psi(2n,x/2), with Psi(n,x) the minimal polynomial of cos(2Pi/n), with rational coefficient array A181875/A181876. There also references and links are found. See especially Watkins and Zeitlin for Psi(n,x).

The zeros of C(n,x), n>=2, are 2*cos(Pi k/n), with k=1,...,n-1 and gcd(k,2n)=1. For n=1 the zero is -2. Alternatively, these zeros are 2*cos(Pi(2l+1)/n), with l=0,...,floor((n-2)/2) and gcd(2l+1,n)=1. For n=1 take l=0.

The first column looks like the differently signed A020513(n),n>=1.

The polynomial for row n=2^m, m>=1, coincides with the row polynomial R(2^(m-1),x) of A127672. See the factorization of these R-polynomials (also known as Chebyshev C-polynomials) given there. - Wolfdieter Lang, Sep 15 2011

From Wolfdieter Lang, Nov 04 2013: (Start)

The norm N(rho(n)) of rho(n) = 2*cos(Pi/n), n >= 1, in the algebraic number field Q(rho(n)) is given by (-1)^delta(n)* C(n, 0), with C(n, x) of degree delta(n) = A055034(n). If N(rho(n)) equals +1 or -1 then 1/rho(n), which is an element of Q(rho(n)), is in fact an integer in this number field. For the 1/rho(n) formula in terms of the C coefficients see A230079. Thus 1/rho(n) is a Q(rho(n))-integer if and only if C(n, 0) is +1 or -1 , and this happens if and only if n is from the set {A230078(k), k >= 2}.

The negation says that, for n a positive integer, 1/rho(n) is not a Q(rho(n))-integer if and only if n is 1 or of the form 2*p^m, m >= 0 and p a prime, which are the numbers of A138929 including 1.

The proof uses for case (i): n = 2*m+1, m >= 1, the fact that C(2*m+1, 0)^2 = (product( 2*cos(Pi*(2*l+1)/(2*m+1)), l=0 .. m-1 and gcd(2*l+1, 2*m+1) = 1))^2 = (product(2*cos(Pi*k/(2*m+1)), k=1..L and gcd(k, 2*m+1) = 1))^2 = cyclotomic(2*m +1, -1). See the linked Q(rho(n)) paper, eq. (31), for a product formula for cyclotomic(n, -1). With the prime factorization of 2*m+1, and the fact that only the squarefree kernel of 2*m+1 enters (see an Oct 29 2013 comment on A013595), one finds, form the formula for cyclotomic(p1*p2*...*pk, x) involving the Moebius function, cyclotomic(2*m +1, -1) = +1, m >= 1. C(1, 0) = +2. For case (ii): n even, one has C(2^m, 0) = 0, -2, +2, for m = 1 , 2, >=3, respectively (see eq. (39) of the linked Q(rho(n)) paper). For odd prime p: (-1)^((p-1)/2)*C(2*p^m, 0) = cyclotomic(2*p^m, -1) = cyclotomic(2*p, -1) = cyclotomic(p, +1) = p, for m >= 1. For more than one odd prime in the squarefree kernel of n = 2*m, m >= 1, one finds C(2*m, 0) = +1 from cyclotomic(2*p1*...*pk, -1) = +1, for k >= 2. (End)

For the conversion of the C-polynomials into sums of Chebyshev's S-polynomials (A049310) see A255237. - Wolfdieter Lang, Mar 16 2015

Examples

			n=1:  2, 1;
n=2:  0, 1;
n=3: -1, 1;
n=4: -2, 0, 1;
n=5: -1,-1, 1;
n=6: -3, 0, 1;
n=7:  1,-2,-1, 1;
n=8:  2, 0,-4, 0, 1;
n=9: -1,-3, 0, 1;
n=10: 5, 0,-5, 0, 1;
...
C(2,x) = R(1,x), C(4,x) = R(2,x), C(8,x) = R(4,x),... with R(n,x) from A127672. - _Wolfdieter Lang_, Sep 15 2011
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10064 (first 220 rows, flattened)
• Wolfdieter Lang, Minimal Polynomials of 2*cos(pi/n)
• Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012-2017.
• Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Crossrefs

Cf. A055034, A181875/A181876, A181877.

Cf. A192003 (row sums). A192004 (alternating row sums).

Programs

Maple

f:= proc(n) local P,z,j;
   P:= factor(evala(Norm(z-convert(2*cos(Pi/n),RootOf))));
   if type(P,`^`) then P:= op(1,P) fi;
   seq(coeff(P,z,j),j=0..degree(P));
end proc:
seq(f(n),n=1..20); # Robert Israel, Aug 04 2015

Mathematica

Flatten[ CoefficientList[ Table[ MinimalPolynomial[2*Cos[Pi/n], x], {n, 1, 17}], x]] (* Jean-François Alcover, Sep 26 2011 *)

PARI

halftot(n)=if(n<=2, 1, eulerphi(n)/2); \\ A023022
default(realprecision, 110);
row(n) = Vecrev(algdep(2*cos(2*Pi/n), halftot(n))); \\ Michel Marcus, Sep 19 2023

Formula

a(n,m) = [x^m] C(n,x), n>=1, m=0..A055034(n), with the minimal (monic and integer) polynomial C(n,x) of 2*cos(Pi/n). See the comment above.