A347929 a(n) = 2^(-1 + (n + n mod 2)/2)*abs(permanent(M_n)) where M_n is the n X n matrix M_n(j, k) = cos(Pi*j*k/n) if n >= 1 and a(0) = 1.

1, 1, 1, 2, 0, 6, 12, 12, 96, 108, 240, 380, 0, 28428, 8176, 16200, 387072, 2817324, 6065280, 4604796, 56832000, 14574168, 2092107072, 13994428360, 8725045248, 162749055000, 1304167707648, 3291435901044, 17899142381568, 107056050266172

Offset: 0

Author

Peter Luschny, Sep 19 2021

Keywords

• nonn

Examples

			a(6) = 16*cos^4(Pi/8) + 8*cos^2(Pi/8) - 64*cos^2(Pi*3/8)*cos^2(Pi/8) + 8*cos^2(Pi*3/8) + 16*cos^4(Pi*3/8).
		

Links

• Zhi-Wei Sun, Fedor Petrov, A surprising identity, discussion in MathOverflow, Jan 17 2019.
• Zhi-Wei Sun, On some determinants involving the tangent function, arXiv:1901.04837 [math.NT], 2021.

Crossrefs

Cf. A347281.

Programs

PARI

p(n) = matpermanent(matrix(n, n, j, k, cos((Pi*j*k)/n)));
A347929(n) = abs(round(2^(-1 + (n + n %2)/2)*p(n)));
{for(n = 0, 12, print(A347929(n)))}

SageMath

def A347929(n):
    if n == 0: return 1
    RF = RealField(100) # adjust precision if needed
    M = matrix(RF, n, n, lambda j, k: cos(j * k * pi / n))
    c = 2^(-1 + (n + n % 2) // 2)
    return abs(round(c*M.permanent()))
print([A347929(n) for n in range(12)])