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%I #12 Jan 30 2016 03:30:18
%S 1,-3,1,5,-5,1,-4,5,-1,9,-30,27,-9,1,-11,55,-77,44,-11,1,4,-13,7,-1,
%T -15,140,-378,450,-275,90,-15,1,17,-204,714,-1122,935,-442,119,-17,1,
%U -4,25,-26,9,-1,0,21,-385,2079,-5148,7007,-5733,2940,-952,189,-21,1,-8,126,-539,967,-870,429,-118,17,-1,0
%N Table of coefficients of the algebraic number s(2*l+1) = 2*sin(Pi/(2*l+1)) as a polynomial in odd powers of rho(2*(2*l+1)) = 2*cos(Pi/(2*(2*l+1))) (reduced version).
%C In the regular (2*l+1)-gon, l >= 1, inscribed in a circle of radius R the length ratio side/R is s(2*l+1) = 2*sin(Pi/(2*l+1)). This can be written as a polynomial in the length ratio (smallest diagonal)/side in the (2*(2*l+1))-gon given by rho(2*(2*l+1)) = 2*cos(Pi/(2*(2*l+1))). This leads, in a first step, to the signed triangle A111125. Because of the minimal polynomial of the algebraic number rho(2*(2*l+1)) of degree delta(2*(2*l+1)) = A055034(2*(2*l+1)), called C(2*(2*l+1),x) (with coefficients given in A187360), one can eliminate all powers rho(2*(2*l+1))^k with k >= delta(2*(2*l+1)) by using C(2*(2*l+1),rho(2*(2*l+1))) = 0. This leads to the present table expressing s(2*(l+1)) in terms of odd powers of rho(2*(2*l+1)) with maximal exponent delta(2*(2*l+1))-1.
%C This table gives the coefficients of s(2*l+1), related to the (2*l+1)-gon, in the power basis of the algebraic number field Q(rho(2*(2*l+1))) of degree delta(2*(2*l+1)), related to rho from the (2*(2*l+1))-gon, provided one inserts zeros for the even powers, starting each row with a zero and filling zeros at the end in order to obtain the row length delta(2*(2*l+1)). Note that some trailing zeros in the present table (e.g., row l = 10) have been given such that the row length for the s(2*l+1) coefficients in the power basis Q(rho(2*(2*l+1))) becomes just twice the one of this table.
%C Thanks go to Seppo Mustonen for telling me about his findings regarding the square of the sum of all length in the regular n-gon, which led me to consider this entry (even though for odd n this is not needed because only s(2*l+1)^2 = 4 - rho(2*l+1)^2 enters).
%F a(l,m) = [x^(2*m+1)](s(2*l+1,x)(mod C(2*(2l+1),x))), with s(2*l+1,x) = sum((-1)^(l-1-s)* A111125(l1,s)*x^(2*s+1), s=0..l-1), l >= 1, m=0, ..., (delta(2*(2*l+1))/2 - 1), with delta(n) = A055034(n).
%F Rows 9,15,21,27 are coefficients of polynomials in reciprocal powers of u for rows n=2,4,6,8 generated by the o.g.f. (u-4)/(u-ux+x^2) of A267633. These polynomials in u occur in a moving average of the polynomials of A140882 interlaced with these polynomials. - _Tom Copeland_, Jan 16 2016
%e The table a(l,m), with n = 2*l+1, begins:
%e n, l \m 0 1 2 3 4 5 6 7 8 9 10
%e 3, 1: 1
%e 5, 2: -3 1
%e 7, 3: 5 -5 1
%e 9, 4: -4 5 -1
%e 11, 5: 9 -30 27 -9 1
%e 13, 6: -11 55 -77 44 -11 1
%e 15, 7: 4 -13 7 -1
%e 17, 8: -15 140 -378 450 -275 90 -15 1
%e 19, 9: 17 -204 714 -1122 935 -442 119 -17 1
%e 21, 10: -4 25 -26 9 -1 0
%e 23, 11: 21 -385 2079 -5148 7007 -5733 2940 -952 189 -21 1
%e 25, 12: -8 126 -539 967 -870 429 -118 17 -1 0
%e 27, 13: 4 -41 70 -43 11 -1 0 0 0
%e ...
%e n = 29 l = 14: -27, 819, -7371, 30888, -72930, 107406, -104652, 69768, -32319, 10395, -2277, 324, -27, 1.
%e n = 5, l=2: s(5) = -3*rho(10) + rho(10)^3 = (tau - 1)*sqrt(2 + tau), approximately 1.175570504, where tau = (1 + sqrt(5))/2 (golden section).
%e n = 17, l = 8: s(17) = -15*x + 140*x^3 - 378*x^5 + 450*x^7 - 275*x^9 + 90*x^11 - 15*x^13 + 1*x^15, with x = rho(34) = 2*cos(Pi/34). s(17) is approximately 0.3674990356. With the length row l = 8 the degree of the algebraic number s(17) = 2*sin(Pi/17) is therefore 2*8 = 16. See A228787 for the decimal expansion of s(17) and A228788 for the one of rho(34).
%Y Cf. A055034, A187360, A228783 (even n case), A228786 (minimal polynomials).
%Y Cf. A140882, A267633.
%K sign,tabf
%O 1,2
%A _Wolfdieter Lang_, Oct 07 2013
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