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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.

%I A347929 #10 Sep 19 2021 13:04:21
%S A347929 1,1,1,2,0,6,12,12,96,108,240,380,0,28428,8176,16200,387072,2817324,
%T A347929 6065280,4604796,56832000,14574168,2092107072,13994428360,8725045248,
%U A347929 162749055000,1304167707648,3291435901044,17899142381568,107056050266172
%N 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.
%H A347929 Zhi-Wei Sun, Fedor Petrov, <a href="https://mathoverflow.net/questions/321098/">A surprising identity</a>, discussion in MathOverflow, Jan 17 2019.
%H A347929 Zhi-Wei Sun, <a href="https://arxiv.org/abs/1901.04837">On some determinants involving the tangent function</a>, arXiv:1901.04837 [math.NT], 2021.
%e A347929 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).
%o A347929 (SageMath)
%o A347929 def A347929(n):
%o A347929     if n == 0: return 1
%o A347929     RF = RealField(100) # adjust precision if needed
%o A347929     M = matrix(RF, n, n, lambda j, k: cos(j * k * pi / n))
%o A347929     c = 2^(-1 + (n + n % 2) // 2)
%o A347929     return abs(round(c*M.permanent()))
%o A347929 print([A347929(n) for n in range(12)])
%o A347929 (PARI)
%o A347929 p(n) = matpermanent(matrix(n, n, j, k, cos((Pi*j*k)/n)));
%o A347929 A347929(n) = abs(round(2^(-1 + (n + n %2)/2)*p(n)));
%o A347929 {for(n = 0, 12, print(A347929(n)))}
%Y A347929 Cf. A347281.
%K A347929 nonn
%O A347929 0,4
%A A347929 _Peter Luschny_, Sep 19 2021