A152191 a(n)=Product_{k=1..floor((n-1)/2)} (1 + 4*cos(k*Pi/n)^2)*(1 - 4*sin(k*Pi/n)^2).

1, 1, 1, -4, -3, 5, 0, 13, 21, -68, -55, 89, 0, 233, 377, -1220, -987, 1597, 0, 4181, 6765, -21892, -17711, 28657, 0, 75025, 121393, -392836, -317811, 514229, 0, 1346269, 2178309, -7049156, -5702887, 9227465, 0, 24157817, 39088169, -126491972, -102334155

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Nov 28 2008

Keywords

• sign

Crossrefs

Cf. A152189.

Programs

Mathematica

f[n_] = Product[(1 + 4*Cos[k*Pi/n]^2)*(1 - 4*Sin[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; Table[N[f[n]], {n, 0, 30}]; Round[%]

PARI

a(n) = round(prod(k=1, floor((n-1)/2), (1+4*cos(k*Pi/n)^2)*(1-4*sin(k*Pi/n)^2))) \\ Colin Barker, Apr 11 2014

Formula

Conjecture: a(2n) = A108196(n-1), n>=2. a(n) = (-1)^(n+1)*A000045(n) *A101675(n-1), n>0. G.f.: 1 -x*(x-1)*(x^2-x+1)*(1+x)^3 / ( (x^4-x^3+2*x^2+x+1)*(x^4+x^3+2*x^2-x+1) ). - R. J. Mathar, Mar 08 2011

Extensions

More terms from Colin Barker, Apr 11 2014