A337939 - OEIS

%I #9 Sep 19 2023 10:22:25

%S 1,1,1,1,0,1,1,0,1,1,0,1,2,1,0,1,-1,0,1,1,0,1,-1,0,1,0,-2,0,1,1,0,1,

%T -1,0,1,1,1,1,0,1,-1,0,1,0,-2,0,1,-4,0,2,1,0,1,-1,0,1,0,-2,0,1,1,0,-3,

%U 0,1,1,0,1,-1,0,1,0,-2,0,1,0,0,1,0,2

%N Irregular triangle T(n, m) read by rows: row n gives the distinct length ratios diagonal/side of regular n-gons, DSR(n, k), for n >= 2, k = 1, 2, ..., floor(n/2), expressed by the coefficients in the power basis of the Galois group Gal(Q(rho(n))/Q), where rho(n) = 2*cos(Pi/n), for n >= 2. T(1, 1) is set to 1.

%C The length of row n is given in A338431(n), for n >= 1.

%C The length of the sublists t(n, k) of the power basis coefficients of DSR(n, k), for k = 1, 2, ..., floor(n/2), is 1 if n = 1, for n >= 2 it is k except for n = n(j) = A111774(j) for which the final A219839(n) sublists have fewer than k members.

%C Trailing vanishing coefficients of the delta(n) = A055034(n) power base elements <1 = rho(n)^0, rho(n)^1, ..., rho(n)^{delta(n)-1}> are not recorded. The coefficients of the minimal polynomial C(n, x) of rho(n) = 2*cos(Pi/n) of degree delta(n) are given in A187360. C(n, rho(n)) = 0 is used to eliminate all powers of rho with exponent >= delta(n).

%C The length ratios DSR(n, k) := diagonal(n, k)/side(n) of regular n-gons, for n >= 2, and k = 1, 2, ..., floor(n/2) (distinct diagonals, starting with the side for k = 1, in increasing order) are given by DSR(n, k) = S(k-1, rho(n)), with the Chebyshev S polynomials (A049310). See the W. Lang link.

%C For n = 2, the degenerate case, diagonal/side = side/side = 1 for k = 1. For n = 1 (a point) diagonal/side is undetermined, and T(1, 1) is set to 1.

%C For the power basis sublists t(n, k), for k = 1, 2, ..., delta(n), only the k coefficients of S(k-1, x) are present (trailing vanishing coefficients are not recorded). For k = delta(n)+1, ..., floor(n/2) less than k coefficients appear due to elimination via C(n, rho(n)) = 0. E.g., for n = 6 with delta(6) = 2 the only coefficient for k = 3 is 2 (coefficient of rho^0). This appears for n = n(j) = A111774(j), because then floor(n/2) - delta(n) = A219839(n) > 0.

%C Because A219839(n) = 0 means that n is from A174090, i.e., a prime or a power of 2 (complement of A111774), these rows n have all the sublists t(n, k) with the k coefficients of S(k-1, x), hence they are identical (but the basis differs). See especially the table for the pairs of consecutive numbers n with identical coefficients, like (2, 3), (4, 5), (16, 17), (256, 267), (65536, 65537), ?... (cf. Fermat primes A019434).

%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>, arXiv:1210.1018 [math.GR], 2012-2017.

%F T(1, 1) = 1, and in row n, for n >= 2, the power base coefficients of Gal(Q(2*cos(Pi/n))/Q) for DSR(n, k) := diagonal(n, k)/side(n) of regular n-gons, for k = 1, 2, ..., floor(n/2), are listed as t(n, k) in this order, with trailing vanishing coefficients omitted.

%e The irregular triangle T(n, m) begins: (For n >= 4 the bar divides the DSR(n, k) power basis coefficients, the sublists t(n, k), for k = 1, 2, ..., floor(n/2))

%e n \ m  1   2 3    4  5  6   7  8 9 10   11 12 12  13 14  15 16 17 18 19 20 ...

%e 1:     1

%e 2:     1

%e 3:     1

%e 4:     1 | 0 1

%e 5:     1 | 0 1

%e 6:     1 | 0 1 |  2

%e 7:     1 | 0 1 | -1  0  1

%e 8:     1 | 0 1 | -1  0  1 | 0 -2 0  1

%e 9:     1 | 0 1 | -1  0  1 | 1  1

%e 10:    1 | 0 1 | -1  0  1 | 0 -2 0  1 | -4  0  2

%e 11:    1 | 0 1 | -1  0  1 | 0 -2 0  1 |  1  0 -3   0  1

%e 12:    1 | 0 1 | -1  0  1 | 0 -2 0  1 |  0  0  1 | 0  2

%e 13:    1 | 0 1 | -1  0  1 | 0 -2 0  1 |  1  0 -3   0  1 | 0  3  0 -4  0  1

%e ...

%e n = 14: 1 | 0 1 | -1 0 1 | 0 -2 0 1 | 1 0 -3 0 1 | 0 3 0 -4 0 1 | 6 0 -8 0 2,

%e n = 15: 1 | 0 1 | -1 0 1 | 0 -2 0 1 | 0 4 1 -1 | 1 -2 0 1 | -1 1 1,

%e n = 16 and n = 17: 1 | 0 1 | -1  0 1 | 0 -2 0 1 | 1 0 -3 0 1 | 0 3 0 -4 0 1 | -1 0 6 0 -5 0 1 | 0 -4 0 10 0 -6 0 1,

%e --------------------------------------------------------------------------------

%e n = 5: DSR(5, 1) = 1 = side(5)/side(5), DSR(5, 2) = 1*rho(5) = A001622 (golden section).

%e n = 8: DSR(8, 1) = 1 = side(8)/side(8), DSR(8, 2) = 1*rho(8) = sqrt(2+sqrt(2)) = A179260, DSR(8, 3) = -1 + rho(8)^2 = 1 + sqrt(2) = A014176, DSR(8, 4) = -2*rho(8) + 1*rho(8)^3 = sqrt(2)*rho(8) = A121601.

%Y Cf. A001622, A055034, A014176, A049310, A019434, A111774, A121601, A179260, A187360, A219839, A338429, A337940, A338431.

%K sign,tabf,easy

%O 1,13

%A _Wolfdieter Lang_, Jan 15 2021