A334431 - OEIS

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

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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