A340183 a(n) = Product_{1<=j,k,m<=n-1} (4*sin(j*Pi/(2*n))^2 + 4*sin(k*Pi/(2*n))^2 + 4*sin(m*Pi/(2*n))^2).

1, 6, 1157625, 170875128460147163136, 448524809573174705684873233798538664686384705625

Offset: 1

Author

Seiichi Manyama, Dec 31 2020

Keywords

• nonn

Comments

(a(n)/(n*3^(n-1)))^(1/3) is an integer.

Crossrefs

Cf. A007341, A124647, A340181, A340182.

Programs

Mathematica

Round[Table[2^((n-1)^3)* Product[3 - Cos[j*Pi/n] - Cos[k*Pi/n] - Cos[m*Pi/n], {j, 1, n-1}, {k, 1, n-1}, {m, 1, n-1}], {n, 1, 5}]] (* Vaclav Kotesovec, Jan 04 2021 *)

PARI

default(realprecision, 500);
{a(n) = round(prod(j=1, n-1, prod(k=1, n-1, prod(m=1, n-1, 4*sin(j*Pi/(2*n))^2+4*sin(k*Pi/(2*n))^2+4*sin(m*Pi/(2*n))^2))))}

Formula

a(n) = Product_{1<=i,j,k<=n-1} (4*f(i*Pi/(2*n))^2 + 4*g(j*Pi/(2*n))^2 + 4*h(k*Pi/(2*n))^2), where f(x), g(x) and h(x) are sin(x) or cos(x).

Limit_{n->infinity} a(n)^(1/n^3) = exp(8*A340322/Pi^3). - Vaclav Kotesovec, Jan 05 2021