A334431 Irregular triangle read by rows: T(m,k) gives the coefficients of x^k of the minimal polynomials of (2*cos(Pi/(2*m)))^2, for m >= 1.

0, 1, -2, 1, -3, 1, 2, -4, 1, 5, -5, 1, 1, -4, 1, -7, 14, -7, 1, 2, -16, 20, -8, 1, -3, 9, -6, 1, 1, -12, 19, -8, 1, -11, 55, -77, 44, -11, 1, 1, -16, 20, -8, 1

Offset

1,3

Comments

The length of row m is delta(m) + 1 = A055034(m) + 1.

For details see A334429, where the formula for the minimal polynomial MPc2(m, x) of 2*cos(Pi/(2*m))^2 = rho(2*m)^2 is given.

The companion triangle for odd n is A334432.

Links

Table of n, a(n) for n=1..44.

Formula

T(m, k) = [x^k] MPc2even(m, x), with MPc2even(m, x) = Product_{j=1..delta(m)} (x - (2 + R(rpnodd(m)_j, rho(m)))) (evaluated using C(m, rho(m)) = 0), for m >= 2, and MPc2even(1, x) = x. Here R(n, x) is the monic Chebyshev R polynomial with coefficients given in A127672. C(n, x) is the minimal polynomial of rho(n) = 2*cos(Pi/n) given in A187360, and rpnodd(m) is the list of positive odd numbers coprime to m and <= m - 1.

Example

The irregular triangle T(m, k) begins:
m,   n \ k  0   1   2    3   4    5   6 ...
-------------------------------------------
1,   2:     0   1
2,   4:    -2   1
3,   6:    -3   1
4,   8:     2  -4   1
5,  10:     5  -5   1
6,  12:     1  -4   1
7,  14:    -7  14  -7    1
8,  16:     2 -16  20   -8   1
9,  18:    -3   9  -6    1
10, 20:     1 -12  19   -8   1
11, 22:   -11  55 -77   44 -11    1
12, 24:     1 -16  20   -8   1
13, 26:    13 -91 182 -156  65  -13   1
14, 28:     1 -24  86 -104  53  -12   1
15, 30:     1  -8  14   -7   1
...

Crossrefs

Cf. A055034, A334429, A334432.

Sequence in context: A319845 A319847 A334217 * A342011 A087295 A175344

Adjacent sequences: A334428 A334429 A334430 * A334432 A334433 A334434

Keyword

sign,tabf,easy

Author

Wolfdieter Lang, Jun 15 2020

Status

approved