A152152 a(n) = Product_{k=1..n} (1 + 4*sin(2*Pi*k/n)^2).

0, 1, 1, 16, 25, 121, 256, 841, 2025, 5776, 14641, 39601, 102400, 271441, 707281, 1860496, 4862025, 12752041, 33362176, 87403801, 228765625, 599074576, 1568239201, 4106118241, 10749542400, 28143753121, 73680216481, 192900153616

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Nov 26 2008

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..2000
• M. Baake, J. Hermisson, and P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See Table 4.
• Kh. Bibak and M. H. Shirdareh Haghighi, Some Trigonometric Identities Involving Fibonacci and Lucas Numbers , Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.4
• N. Garnier and O. Ramaré, Fibonacci numbers and trigonometric identities, April 2006.
• N. Garnier and O. Ramaré, Fibonacci numbers and trigonometric identities, Fibonacci Quart. 46/47 (2008/2009), no. 1, 56-61.

Crossrefs

Cf. A000032, A001350, A324487.

Programs

Magma

[(1-Lucas(n)+(-1)^n)^2: n in [0..30]]; // G. C. Greubel, Mar 13 2019

Mathematica

Table[(1 + Fibonacci[n] - 2*Fibonacci[n+1] + (-1)^n)^2, {n, 0, 30}]

PARI

{a(n) = (1-fibonacci(n-1)-fibonacci(n+1)+(-1)^n)^2}; \\ G. C. Greubel, Mar 13 2019

Sage

[(1-lucas_number2(n,1,-1)+(-1)^n)^2 for n in (0..30)] # G. C. Greubel, Mar 13 2019

Formula

a(n) = Product_{k=1..n} (1 + 4*sin(2*Pi*k/n)^2).

a(n) = (1 + Fibonacci(n) - 2*Fibonacci(n + 1) + (-1)^n)^2.

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

a(n) = A001350(n)^2. - Colin Barker, Apr 13 2014

a(n) = (1 + (-1)^n - Lucas(n))^2. - G. C. Greubel, Mar 13 2019