A187360 - OEIS

%I #55 Sep 19 2023 12:21:50

%S 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,

%T 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,

%U 1,1,-4,-4,1,1,2,0,-16,0,20,0,-8,0,1,1,4,-10,-10,15,6,-7,-1,1

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

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

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

%C   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).

%C 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.

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

%C 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

%C From _Wolfdieter Lang_, Nov 04 2013: (Start)

%C 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}.

%C 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.

%C 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)

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

%H Robert Israel, <a href="/A187360/b187360.txt">Table of n, a(n) for n = 1..10064</a> (first 220 rows, flattened)

%H Wolfdieter Lang, <a href="/A187360/a187360.pdf">Minimal Polynomials of 2*cos(pi/n)</a>

%H Wolfdieter Lang, <a href="http://arxiv.org/abs/1210.1018">The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon</a>, arXiv:1210.1018 [math.GR], 2012-2017.

%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] 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.

%e n=1:  2, 1;

%e n=2:  0, 1;

%e n=3: -1, 1;

%e n=4: -2, 0, 1;

%e n=5: -1,-1, 1;

%e n=6: -3, 0, 1;

%e n=7:  1,-2,-1, 1;

%e n=8:  2, 0,-4, 0, 1;

%e n=9: -1,-3, 0, 1;

%e n=10: 5, 0,-5, 0, 1;

%e ...

%e 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

%p f:= proc(n) local P,z,j;

%p    P:= factor(evala(Norm(z-convert(2*cos(Pi/n),RootOf))));

%p    if type(P,`^`) then P:= op(1,P) fi;

%p    seq(coeff(P,z,j),j=0..degree(P));

%p end proc:

%p seq(f(n),n=1..20); # _Robert Israel_, Aug 04 2015

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

%o (PARI) halftot(n)=if(n<=2, 1, eulerphi(n)/2); \\ A023022

%o default(realprecision, 110);

%o row(n) = Vecrev(algdep(2*cos(2*Pi/n), halftot(n))); \\ _Michel Marcus_, Sep 19 2023

%Y Cf. A055034, A181875/A181876, A181877.

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

%K sign,easy,tabf

%O 1,1

%A _Wolfdieter Lang_, Jul 14 2011