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 A228780 Power basis components of the algebraic numbers S2(n) in Q(2*cos(Pi/n)), where S2(n) is the square of the sum of the lengths of the distinct line segments (side and diagonals) in the regular n-gon. 4

%I

%S 4,3,6,4,3,4,12,6,-1,4,4,-2,-4,6,4,3,8,4,-16,-8,12,6,3,-4,-8,4,4,0,4,

%T 10,4,3,8,-8,-12,4,4,28,14,-40,-20,12,6,-1,8,12,4,-2,-8,28,28,-26,-20,

%U 6,4,-1,-8,16,28,-16,-20,4,4,4,2,-12,-6,8,4,3,-8,-24,28,44,-20,-24,4,4,0,8,28,-4,-40,-12,10,4,-1,-16,-24,0,12,4

%N Power basis components of the algebraic numbers S2(n) in Q(2*cos(Pi/n)), where S2(n) is the square of the sum of the lengths of the distinct line segments (side and diagonals) in the regular n-gon.

%C The length of row n of this irregular array is the degree of the algebraic number rho(n):= 2*cos(Pi/n), given in A055034(n). See a Jul 19 2011 comment there.

%C The regular n-gon, inscribed in a circle of radius defining the length unit 1, has distinct line segments (chords) (V_0, V_j), j=1, ... , floor(n/2), with the n-gon vertices V_j, j=0, ... , n-1 distributed on the circle in the counterclockwise sense. The corresponding length ratios are denoted by L(n,j)/radius. The side length is s(n) = (V_0, V_1) = 2*sin(Pi/n), and for n >= 4 the first (the smallest) diagonal has length s(n)*rho(n), with rho(n) of degree delta(n) given above. s(2) = 2 is the ratio of the diameter of the circle. rho(2) = 0, but we use here rho(2)^0 = 1.

%C For n = 3: rho(3) = 1, s(3)^2 = 3. The algebraic number field Q(rho(n)) is the subject of the W. Lang link given below.

%C S2(n) := (sum(L(n,j)/radius, j=1, ... ,floor(n/2))^2 is seen below to be a number in the field Q(rho(n)) of degree delta(n), namely S2(n) = sum(a(n,k)*rho(n)^k, k=0..(delta(n)-1)). From the definition one has S2(n) = (s(n)*sum(S(j-1,rho(n)), j=1..floor(n/2)))^2, with the Chebyshev S-polynomials (see A049310). Due to s(n) = s(2*n)*rho(2*n), rho(2*n) = sqrt(2 + rho(n)) and an S-identity this becomes S2(n) = (s(2*n)*S(floor(n/2)-1, rho(2*n))*S(floor(n/2), rho(2*n)))^2. This can also be written as S2(n) = 4*(1 - T(2*floor(n/2), rho(2*n)/2))*(1 - T(2*(floor(n/2)+1), rho(2*n)/2))/(4-rho(2*n)^2), with Chebyshev's T-polynomials (see A053120). S2(n), written as a function of rho(n), has to be computed modulo the minimal polynomial C(n,rho(n)) of degree delta(n). These minimal polynomials are treated in A187360 (see the link to a Galois paper there, with its Table 2 and Section 3). The result is then the above given representation of S2(n) in the power basis of Q(rho(n)).

%C This computation was inspired by an email exchange with Seppo Mustonen. The author thanks him for sending the paper given as a link below. In this connection one should consider the even and odd n cases separately in order to find the square of the total length segments/radius in the regular n-gon, noticing that in the odd n case each distinct chord (side or diagonal) appears 2*(n/2) = n times, whereas in the even n case the longest diagonal of length 2 (in units of the radius) appears only n/2 times and the other chords appear n times.

%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>.

%H Seppo Mustonen, <a href="http://www.survo.fi/papers/Roots2013.pdf"> Lengths of edges and diagonals and sums of them in regular polygons as roots of algebraic equations</a>.

%H Seppo Mustonen, <a href="/A108716/a108716.pdf">Lengths of edges and diagonals and sums of them in regular polygons as roots of algebraic equations</a> [Local copy]

%F a(n,k) = [rho^k] (S2(n) modulo C(n,rho(n)), with S2(n) the square of the sum of the distinct length/radius ratios in the regular n-gon, with rho(n) = 2*cos(Pi/n) given above in a comment, and C(n,x) the minimal polynomial of rho(n) given in A187360 (see Table 2 and section 3 of the paper given in the W. Lang link below).

%e The table a(n,k) begins:

%e n\k 0 1 2 3 4 5 ...

%e 2: 4

%e 3: 3

%e 4: 6 4

%e 5: 3 4

%e 6: 12 6

%e 7: -1 4 4

%e 8: -2 -4 6 4

%e 9: 3 8 4

%e 10: -16 -8 12 6

%e 11: 3 -4 -8 4 4

%e 12: 0 4 10 4

%e 13: 3 8 -8 -12 4 4

%e 14: 28 14 -40 -20 12 6

%e 15: -1 8 2 4

%e ...

%e n=5: S2(5) = (4-rho(5)^2)*(sum(S(j-1,rho(5)), j=1..2))^2 =

%e 4 + 8*rho(5) + 3*rho(5)^2 - 2*rho(5)^3 - rho(5)^4, reduced with C(5,x) =x^2 -x -1, with x = rho(5), using C(5,rho(5)=0, to eliminate all powers of rho(5) starting with power 2.

%e This leads to S2(5) = 3*1 + 4*rho(5). rho(5) = phi, the golden section.

%e The exact or approximate real values for S2(n) are, for n = 2, ..., 15: 4, 3, 11.65685426, 9.472135960, 22.39230484, 19.19566936, 36.32882142, 32.16343753, 53.49096128, 48.37415020, 73.88698896, 67.82742928, 97.52047276, 90.52313112.

%Y Cf. A228781, A228782 (minimal polynomials for odd and even n).

%K sign,tabf

%O 2,1

%A _Wolfdieter Lang_, Oct 01 2013

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