cp's OEIS Frontend

All terms are even.

All these cases are so-called reversed cases when degree of minimal polynomial of cos(Pi/n) = 2*degree of minimal polynomial of sin(Pi/n) (in rest of cases is vice versa).

For k = 1, A055035(1) = 1 is also odd. - Wolfdieter Lang, Nov 06 2019

The elements of the set {k == 2 (mod 4): k = 2 or phi(k/2)/2 is odd} sorted increasingly, where phi = A000010 (Euler's totient). - Wolfdieter Lang, Nov 06 2019

The degree formula given in the I. Niven reference on p. 37-8 (see below) appears as part of theorem 3.9 attributed to D. H. Lehmer. However, this part, concerning sin(2*Pi/n), differs from Lehmer's result, which in fact is incorrect. - Wolfdieter Lang, Jan 09 2011

This is also the algebraic degree of the area of a regular n-gon inscribed in the unit circle. - Jack W Grahl, Jan 10 2011

Every degree appears in this sequence except for the half-nontotients, A079695. - T. D. Noe, Jan 12 2011

See A181872/A181873 for the monic rational minimal polynomial of sin(2*Pi/n), and A181871 for the non-monic integer version. In A231188 the (monic and integer) minimal polynomials for 2*sin(2*Pi/n) are given. - Wolfdieter Lang, Nov 30 2013

The old name was A196445(n)/2.

From Wolfdieter Lang, Nov 12 2019: (Start)

These are the odd numbers m for which the degree of the algebraic number sin(Pi/(2*m)) (the degree of its minimal polynomial), given by A055035(2*m), is odd. Because A055035(1) = 1, there is just this other instance with odd A055035.

This sequence {a(n)} consists for n >= 2 of all powers >= 1 of each prime p == 3 (mod 4) from A002145, sorted into increasing order. This follows from the factorization of odd m >= 3, and that phi(m)/2 has to be odd.

For a(n), with n >= 2, there is exactly one pair of solutions x = +1 and -1 (the trivial solution) of the congruence x^2 == +1 (mod a(n)), and there is no solution of the congruence x^2 == -1 (mod a(n)). The proof starts with showing this for p == 3 (mod 4). It can be shown that the square 1 appears only for x = 1 if x runs through 1, ..., (p-1)/2. The other x range (p+1)/2, ..., p-1, which has the same squares (mod p), can, by reading backwards, be interpreted as the -x partners of x from the first range. The Legendre symbol (-1, p) = -1 shows the second claim. Then one applies the lifting theorem for powers of primes (see Apostol, Theorem 5.30, p. 121), where only its part (a) is needed, and the step by step lifting to each prime power is unique.

For a(1) = 1 there is just one solution x = 0 of the congruence x^2 == +1 (mod 1), and also of x^2 == -1 (mod 1).

The complementary sequence with odd m >= 3 and even phi(m)/2 is given in A327922.

(End)

s(n) := 2*sin(Pi/n) is, for n >= 2, the length ratio side/R of the regular n-gon inscribed in a circle of radius R. This algebraic number s(n), n >= 1, has the degree gamma(n) := A055035(n), and the row length of this table is gamma(n) + 1.

s(n) has been given in the power basis of the relevant algebraic number field in A228783 for even n (bisected into n == 0 (mod 4) and n == 2 (mod 4)), and in A228785 for odd n.

For the computation of the minimal polynomials ps(n,x), using the coefficients of s(n) in the relevant number field, and the conjugates of the corresponding algebraic numbers rho (giving the length ratios (smallest diagonal)/side in the relevant regular polygons see a comment on A228781. Note that the product of the gamma(n) linear factors (x - conjugates) has to be computed modulo the minimal polynomial of the relevant rho(k) = 2*cos(Pi/k) (called C(k,x=rho(k)) in A187360).

Thanks go to Seppo Mustonen, who asked a question about the square of the sum of all lengths in the regular n-gon, which led to this computation of s(n) and its minimal polynomial.

It would be interesting to find out which length ratios in the regular n-gon give the other positive zeros of the minimal polynomial ps(n,x). See some examples below.

The zeros of the row polynomials ps(n,x) are 2*cos(2*Pi*k/c(2*n))) for gcd(k, c(2*n)) = 1, where c(n) = A178182(n), and k from {0, ..., floor(c(2*n)/2)}, for n >= 1. The number of these solutions is gamma(n) = A055035(n). See the formula section. This results from the zeros of the minimal polynomials of sin(2*Pi/n), with coefficients given in A181872/A181873. - Wolfdieter Lang, Oct 30 2019

In the regular (2*l)-gon inscribed in a circle of radius R the length ratio side/R is s(2*l) = 2*sin(Pi/(2*l)). This can be written as a polynomial in the length ratio (smallest diagonal)/side given by rho(2*l) = 2*cos(Pi/(2*l)). (For the 2-gon there is no such diagonal and rho(2) = 0). This leads, in a first step, to the triangle A127672 (see the Oct 05 2013 comment there referring also to the bisections signed A111125 and A127677). Because the minimal polynomial of the algebraic number rho(2*l) of degree delta(2*l) = A055034(2*l), called C(2*l,x) (with coefficients given in A187360) one can eliminate all powers rho(2*l)^k with k >= delta(2*l) by using C(2*l,rho(2*l)) = 0. Furthermore, because for odd l only even powers of rho(2*l) appear, but rho(2*l)^2 = 2 + rho(l), one will obtain a reduced table for s(2*l) with powers rho(2*l)^(2*k+1), k= 0, ..., (delta(2*l)-2)/2 if l is even, and with powers rho(l)^m, m=0, ... , delta(l)-1 if l is odd.

This leads to a reduction of the triangle A127672, when applied for the s(2*l) computation, giving the present table with row length delta(4*L) = A055034(4*L) = phi(8*L)/2 if l =2*L, if L >= 1, and phi(2*L+1)/2 = A055035(4*L+2), if l = 2*L + 1, L >= 1, where phi(n) = A000010(n) (Euler totient).

This table gives the coefficients of s(2*l) in the power basis of the algebraic number field Q(rho(2*l)) of degree delta(2*l) = A055034(2*l) if l is even, and in Q(rho(l)) of degree delta(2*l)/2 if l is odd. s(2) and s(6) are rational integers of degree 1.

Thanks go to Seppo Mustonen whose question about the square of the sum of all length in a regular n-gon, led me to this computation.

If l = 2*L+1, L >= 0, that is n == 2 (mod 4), one can obtain s(2*l) more directly in powers of rho(l) by s(2*l) = R(l-1, rho(l)) (mod C(l,rho(l))), with the monic (except for l=1) Chebyshev T-polynomials, called R, in A127672, and the C polynomials from A187360. - Wolfdieter Lang, Oct 10 2013

Also the algebraic degrees of the surface area and volume of an n-prism with unit edge lengths.

The minimal polynomials of the algebraic number s(n) = 2*sin(Pi/n) of degree gamma(n) = A055035(n) = A093819(2*n) has the zeros 2*cos(2*Pi*T(n,m)/c(2*n)), with c(2*n) = A178182(2*n), for m = 1, 2, ..., gamma(n) and n >= 1.

The number s(n) is the length ratio side(n)/R of the regular n-gon inscribed in a circle of radius R.

The motivation to look at these zeros came from the book of Carl Schick, and the paper by Brändli and Beyne. There, only length ratios diagonals/R in 2*(2*m + 1)-gons, for m >= 1, are considered.

If one is interested in length ratios diagonals/side then the minimal polynomials of rho(n) := 2*cos(Pi/n) (smallest diagonal/side) are important. These are given in A187360, called there C(n, x).