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

0,4

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.

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

Mathematica

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

Prog

(PARI) {a(n) = (1-fibonacci(n-1)-fibonacci(n+1)+(-1)^n)^2}; \\ G. C. Greubel, Mar 13 2019
(Magma) [(1-Lucas(n)+(-1)^n)^2: n in [0..30]]; // 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

Crossrefs

Cf. A000032, A001350, A324487.

Sequence in context: A112392 A109685 A188826 * A260047 A061101 A166672

Adjacent sequences: A152149 A152150 A152151 * A152153 A152154 A152155

Keyword

nonn

Author

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

Status

approved