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%I #39 Jan 09 2024 08:49:21
%S 0,1,1,16,25,121,256,841,2025,5776,14641,39601,102400,271441,707281,
%T 1860496,4862025,12752041,33362176,87403801,228765625,599074576,
%U 1568239201,4106118241,10749542400,28143753121,73680216481,192900153616
%N a(n) = Product_{k=1..n} (1 + 4*sin(2*Pi*k/n)^2).
%H Seiichi Manyama, <a href="/A152152/b152152.txt">Table of n, a(n) for n = 0..2000</a>
%H M. Baake, J. Hermisson, and P. Pleasants, <a href="http://dx.doi.org/10.1088/0305-4470/30/9/016">The torus parametrization of quasiperiodic LI-classes</a>, J. Phys. A 30 (1997), no. 9, 3029-3056. See Table 4.
%H Kh. Bibak and M. H. Shirdareh Haghighi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Bibak/bibak4.html">Some Trigonometric Identities Involving Fibonacci and Lucas Numbers </a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.4
%H N. Garnier and O. Ramaré, <a href="https://ramare-olivier.github.io/Maths/CirculantFibonacci.pdf">Fibonacci numbers and trigonometric identities</a>, April 2006.
%H N. Garnier and O. Ramaré, <a href="http://www.fq.math.ca/Papers1/46_47-1/Ramare_Garnier_11-08.pdf">Fibonacci numbers and trigonometric identities</a>, Fibonacci Quart. 46/47 (2008/2009), no. 1, 56-61.
%F a(n) = Product_{k=1..n} (1 + 4*sin(2*Pi*k/n)^2).
%F a(n) = (1 + Fibonacci(n) - 2*Fibonacci(n + 1) + (-1)^n)^2.
%F G.f.: -x*(x^6 -2*x^5 +10*x^4 -14*x^3 +10*x^2 -2*x +1)/((x -1)*(x +1)*(x^2 -3*x +1)*(x^2 -x -1)*(x^2 +x -1)). - _Colin Barker_, Apr 13 2014
%F a(n) = A001350(n)^2. - _Colin Barker_, Apr 13 2014
%F a(n) = (1 + (-1)^n - Lucas(n))^2. - _G. C. Greubel_, Mar 13 2019
%t Table[(1 + Fibonacci[n] - 2*Fibonacci[n+1] + (-1)^n)^2, {n, 0, 30}]
%o (PARI) {a(n) = (1-fibonacci(n-1)-fibonacci(n+1)+(-1)^n)^2}; \\ _G. C. Greubel_, Mar 13 2019
%o (Magma) [(1-Lucas(n)+(-1)^n)^2: n in [0..30]]; // _G. C. Greubel_, Mar 13 2019
%o (Sage) [(1-lucas_number2(n,1,-1)+(-1)^n)^2 for n in (0..30)] # _G. C. Greubel_, Mar 13 2019
%Y Cf. A000032, A001350, A324487.
%K nonn
%O 0,4
%A _Roger L. Bagula_ and _Gary W. Adamson_, Nov 26 2008
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