cp's OEIS Frontend

This is A118825(n+5), n >= 0. (The g.f. given there is not quite correct, because then a(0) = 0 but it should be 1.)

This sequence a(n), n >= 2, is product(2*cos((2*l+1)*Pi/2), l=0..floor((n-2)/2)). This is the unrestricted product which appears in the formula for C(n, 0) with the minimal polynomial C of rho(n):=2*cos(Pi/n) (see A187360), the length ratio (smallest diagonal)/side in the regular n-gon. The restriction gcd(2*l+1, n) = 1 is ineffective for n = 2^k, k>=1, and for n = odd prime p. Therefore norm(rho(n)) = (-1)^delta(n)*C(n, 0) with delta(n) (see A055034) the degree of C, can be computed from the present sequence for these two cases.

The length of row l of this table is delta(l) + 1 = A055034(l) + 1, l >= 1, that is: 2, 2, 2, 3, 3, 3, 4, 5, 4, 5, 6, 5, 7, 7, 5, ...

s(n):= 2*sin(Pi/n) is the length ratio side/R of a regular n-gon inscribed in a circle of radius R (in some length units). In general s(n)^2 = 4 - rho(n)^2 with rho(n):= 2*cos(Pi/n), the length ratio (smallest diagonal)/s(n) in the regular n-gon (n>=2). If n is even, say 2*l, l>=1, then s(2*l)^2 = 2 - rho(l) (because rho(2*l)^2 = rho(l) + 2). Therefore s(2*l) is an integer in the algebraic number field Q(rho(l)).

Its (monic) minimal polynomial is obtained from the conjugates of rho(l), called rho(l;j), j = 1, 2, ..., delta(l), which are the zeros of the minimal polynomial of rho(l) = rho(l;1) of degree delta(l) = A055034(l), called C(l, x) in A187360. These conjugates are therefore rho(l;j) = 2*cos(Pi*rpnodd(l,j)/l) where rpnodd(l,j) is the j-th entry of the list rpnodd(l) of the odd numbers < l which are relatively prime to l (for example, rpnodd(9) = [1,5,7], and rpnodd(9,2) = 5). From this the conjugates of s(2*l)^2 become 2 - rho(l;j), and the minimal polynomial of s(2*l)^2 is MPs2(l, x) = product( x - (2- rho(l;j)), j=1..delta(l)) for l >=1. Because the zeros of C(l, x) are integers in the algebraic number field Q(rho(l)) written in its power basis (see table 4 of the link under A187360 to the Q(2 cos(Pi/n)) paper) one finds, after expansion and reducing powers of rho(l) modulo C(l, rho(l)), directly the integer coefficients appropriate for this (monic) minimal polynomial. Only the equation C(l, rho(l)) = 0 is needed, not the trigonometric version of rho(l) and its powers.

This computation was motivated by a preprint of S. Mustonen, P. Haukkanen and J. K. Merikoski, called 'Polynomials associated with squared diagonals of regular polygons', Nov 16 2013.

The row length sequence of this table is delta(2*k+1), with the degree delta(n) = A055034(n) of the algebraic number rho(n):= 2*cos(Pi/n), k >= 1.

The numbers S2(n) have been given in A228780 in the power basis of the degree delta(n) number field Q(rho(n)), with rho(n):= 2*cos(Pi/n), n >= 2. Here the odd n case, n = 2*k + 1 is considered. S2(n) is the square of the sum of the distinct length ratios side/radius or diagonal/radius with the radius of the circle in which a regular n-gon is inscribed. For two formulas for S2(n) in terms of powers of rho(n) see the comment section of A228780.

The minimal (monic) polynomial of S2(2*k+1) has degree delta(2*k+1) and is given by

p(2*k+1,x) = Product_{j=1..delta(2*k+1)} (x - S2(2*k+1)^{(j-1)} (mod C(2*k+1,delta(n))) = sum(a(k, m)*x^m, m = 0..delta(2*k+1)), where S2(2*k+1)^{(0)} = S2(2*k+1) and S2(2*k+1)^{(j-1)} is the (j-1)-th conjugate of S2(2*k+1). The conjugate of a number alpha(n) = Sum_{j=0..(delta(n)-1)} b(n, j)*rho(n)^j in Q(rho(n)) is obtained from the conjugates of rho(n), given in turn by the zeros x(n, j) of the minimal polynomial C(n, x) (see A187360 and the link to the W. Lang Galois paper, tables 2 and 3) as rho(n)^{(j-1)} = x(n, j), j = 1..delta(n), with rho(n)^{(0)} = rho(n).

The motivation to look into this problem originated from emails by Seppo Mustonen, who found experimentally polynomials which had as one zero the square of the total length/radius of all chords (sides and diagonals) in the regular n-gon. See his paper given as a link below. The author thanks Seppo Mustonen for sending his paper.

If the minimal polynomial of the algebraic number S2(n) in the n-gon with n = 2*k+1 is p(n, x) then the minimal polynomial of the square of the sum of the length of all n sides and n*(n-3)/2 diagonals is P(n, x) = n^(2*delta(n))*p(n, x/n^2).

rho(34):= 2*cos(Pi/34) is used in the algebraic number field Q(rho(34)) of degree 16 (see A187360) in which s(17) = 2*cos(Pi/17) (for its decimal expansion see A228787), the length ratio side/R of a regular 17-gon inscribed in a circle of radius R, is an integer. See A228787 for this expansion.

Gauss' formula for cos(2*Pi/17), given in A210644, can be inserted into rho(34) = sqrt(2+sqrt(2+2*cos(2*Pi/17))).

The minimal polynomial of rho(34) is 17 - 204*x^2 + 714*x^4 - 1122*x^6 + 935*x^8 - 442*x^10 + 119*x^12 - 17*x^14 + x^16 (row n=34 polynomial of A187360).

The continued fraction expansion starts with 1; 1, 116, 4, 1, 2, 1, 20, 2, 2, 1, 7, 10, 2, 2, 1, 3, 6, 1, 4, 4, 15, ...

The length of row n is delta(A230078(n)), n>=2, with delta(k) = A055034(k).

The power base of the algebraic number field Q(rho(k)), with rho(k):= 2*cos(Pi/k), k >= 2, is <1, rho(k), rho(k)^2, ..., rho(k)^(delta(k)-1)>. Q(rho(k))-integers have integer coefficients in this basis. A230078(n), n >= 2, gives precisely the numbers k for which the inverse 1/rho(k) is a Q(rho(k))-integer. The present table a(n,m) lists these integer coefficients for 1/rho(A230078(n)), n >= 2, m = 0, 1, ..., delta(A230078(n))-1. delta(k) is the degree of the minimal polynomial C(k, x) of rho(k) (see A187360).

In general, 1/rho(k) = -(sum(c(k, m+1)*rho(k)^m, m=0..delta(k)-1))/c(k, 0), k >= 2, with the coefficients c(k ,m) of the minimal polynomial C(k, x) given in A187360(k, m). c(k ,0) = C(k, x=0) is +1 or -1 if and only if k is from {A230078(n), n>=2}, leading to a Q(rho(k))-integer.

omega(11):= S(4, x) = 1 - 3*x^2 + x^4 with x = rho(11) := 2*cos(Pi/11). See A049310 for Chebyshev s-polynomials. rho(11) is the ratio (shortest diagonal)/side in a regular 11-gon. See the Q(2*cos(Pi/n)) link given in A187360. This is the power basis representation of omega(11) in the algebraic number field Q(2*cos(Pi/11)) of degree 5.

omega(11) = 1/(2*cos(Pi*5/11)) = 1/R(5, rho(11)) with the R-polynomial given in A127672. This follows from a computation of the power basis coefficients of the reciprocal of R(5, x) (mod C(11, x)) = 1+2*x-3*x^2-x^3+x^4, where C(11, x) is the minimal polynomial of rho(11) given in A187360. The result for this reciprocal (mod C(11, x)) is 1 - 3*x^2 + x^4 giving the power base coefficients [1,0,-3,0,1] for omega(11).

omega(11) is the analog in the regular 11-gon of the golden section in the regular 5-gon (pentagon), because it is the limit of a(n+1)/a(n) for n -> infinity of sequences like A038342, A069006, A230080 and A230081.

The ratio diagonal/side of the second and third shortest diagonals in a regular 11-gon are respectively x^2 - 1 and x^3 - 2*x, where x = 2*cos(Pi/11). - Mohammed Yaseen, Nov 03 2020

The length of row l is delta(2*l+1) + 1 = A055034(2*l+1) + 1, l >= 0.

See the comments on A232631 (even n case) for s(n) = 2*sin(Pi/n) and the minimal polynomial of s(n)^2. Here n = 2*l+1 and s(2*l+1)^2 = 4 - rho(2*l+1)^2 is an integer in the algebraic number field Q(rho(2*l+1)). The minimal polynomial of s(2*l+1)^2 is then MPs2(2*l+1, x) = product(x - 2*(1 + cos(Pi*rpnodd(2*l+1,j)/(2*l+1))), j=1..delta(2*l+1)), l >= 0, where rpnodd(2*l+1) is the list of positive odd numbers < 2*l+1 and relatively prime to 2*l+1. rpnodd(2*l+1,j) is the j-th member of this increasingly ordered list. Here the identity 4 - (2*cos(Pi*(2*k+1)/(2*l+1)))^2 = 2*(1 - cos(Pi*2*(2*k+1)/(2*l+1))) = 2*(1 - (- cos(Pi*(2*l+1 - 2*(2*k+1))/ (2*l+1)))) has been used, and for 2*k+1 < 2*l+1 and gcd(2*k+1, 2*l+1) = 1 this becomes the product given above because 1 = gcd(-(2*k+1), 2*l+1) = gcd(-2*(2*k+1), 2*l+1) = gcd(2*l+1, -2*(2*k+1) + (2*l+1)).

This computation was motivated by a preprint of S. Mustonen, P. Haukkanen and J. K. Merikoski, called ``Polynomials associated with squared diagonals of regular polygons'', Nov 16 2013.

a(n) is obtained from the Girard-Waring formula for the sum of powers of N = 4 indeterminates (see A324602), with the elementary symmetric functions e_1 = -1, e_2 = -4, e_3 = -4 and e_4 = 1. The arguments are e_j(x_1, x_2, x_3, x_4), for j = 1..4, with the zeros {x_i}A187360,%20for%20n%20=%2015),%20appearing%20to%20the%20power%20n%20in%20the%20formula%20given%20above.%20-%20_Wolfdieter%20Lang">{i=1..4} of the minimal polynomial of 2*cos(Pi/15) (see A187360, for n = 15), appearing to the power n in the formula given above. - _Wolfdieter Lang, May 08 2019

a(n) = A055034(A007520(n)), n >= 1. This is the degree of the minimal polynomial C(A007520(n) ,x) of 2*cos(Pi/A007520(n)) (see A187360). a(n) is of course congruent 1 (mod 4). - Wolfdieter Lang, Oct 24 2013

The coefficients of the minimal polynomials C(N,x) of the algebraic number 2*cos(Pi/N) are given in A187360, where N is n.

The degree of C(N,x) is delta(N) = 1 if N=1 and delta(N) = phi(2*N)/2 if N > 1, with Euler's totient function phi(n) = A000010(n).

The forbidden degree values are shown in the complement (relative to the positive integers) A079695.

The array of the values N (the indices) for which the degree delta(N) = a(n), n >= 1, is given in A207334.