cp's OEIS Frontend

A228785 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).

%I A228785 #12 Jan 30 2016 03:30:18
%S A228785 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 A228785 -15,140,-378,450,-275,90,-15,1,17,-204,714,-1122,935,-442,119,-17,1,
%U A228785 -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 A228785 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 A228785 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 A228785 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 A228785 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 A228785 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 A228785 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 A228785 The table a(l,m), with n = 2*l+1, begins:
%e A228785 n,   l \m  0    1     2     3    4     5    6    7   8   9 10
%e A228785 3,   1:    1
%e A228785 5,   2:   -3    1
%e A228785 7,   3:    5   -5     1
%e A228785 9,   4:   -4    5    -1
%e A228785 11,  5:    9  -30    27    -9    1
%e A228785 13,  6:  -11   55   -77    44  -11     1
%e A228785 15,  7:    4  -13     7    -1
%e A228785 17,  8:  -15  140  -378   450 -275    90  -15    1
%e A228785 19,  9:   17 -204   714 -1122  935  -442  119  -17   1
%e A228785 21, 10:   -4   25   -26     9   -1     0
%e A228785 23, 11:   21 -385  2079 -5148 7007 -5733 2940 -952 189 -21  1
%e A228785 25, 12:   -8  126  -539   967 -870   429 -118   17  -1   0
%e A228785 27, 13:    4  -41    70   -43   11    -1    0    0   0
%e A228785 ...
%e A228785 n = 29 l =  14:  -27, 819, -7371, 30888, -72930, 107406, -104652, 69768, -32319, 10395, -2277, 324, -27, 1.
%e A228785 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 A228785 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 A228785 Cf. A055034, A187360, A228783 (even n case), A228786 (minimal polynomials).
%Y A228785 Cf. A140882, A267633.
%K A228785 sign,tabf
%O A228785 1,2
%A A228785 _Wolfdieter Lang_, Oct 07 2013