A152189 Product_{k=1..floor((n-1)/2)} (1 + 4*cos(k*Pi/n)^2)*(1 + 4*sin(k*Pi/n)^2).

1, 1, 1, 8, 9, 55, 64, 377, 441, 2584, 3025, 17711, 20736, 121393, 142129, 832040, 974169, 5702887, 6677056, 39088169, 45765225, 267914296, 313679521, 1836311903, 2149991424, 12586269025, 14736260449, 86267571272, 101003831721, 591286729879, 692290561600

Offset: 0

Author

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

Keywords

• nonn

Comments

It appears that limit(sqrt(a(n+2)/a(n)), n->Infinity) = 1+(sqrt(5)+1)/2.

Crossrefs

Cf. A152191.

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[%] (* corrected by Colin Barker, Apr 11 2014 *)

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

Empirical g.f.: (x^6+x^5-9*x^4+7*x^2-x-1) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+3*x+1)). - Colin Barker, Apr 11 2014

Extensions

Two initial terms added, and several terms corrected by Colin Barker, Apr 11 2014