A340322 Decimal expansion of Integral_{x=0..Pi/2, y=0..Pi/2, z=0..Pi/2} log(4*cos(x)^2 + 4*cos(y)^2 + 4*cos(z)^2) dz dy dx.

6, 4, 8, 5, 6, 9, 6, 4, 6, 5, 2, 1, 8, 4, 9, 7, 6, 9, 3, 7, 0, 8, 5, 8, 1, 3, 7, 2, 1, 0, 3, 3, 1, 5, 7, 6, 4, 1, 5, 2, 2, 6, 6, 3, 2, 5, 6, 1, 7, 9, 7, 6, 3, 1, 6, 8, 3, 1, 7, 3, 8, 8, 4, 2, 4, 5, 2, 5, 5, 5, 2, 3, 8, 7, 8, 4, 0, 0, 5, 6, 7, 8, 5, 4, 1, 8, 8, 8, 8, 7, 6, 4, 1, 9, 6, 8, 5, 7, 5, 5, 3, 9, 1, 7, 4

Offset: 1

Author

Vaclav Kotesovec, Jan 04 2021

Keywords

• nonn
• cons

Comments

Integral_{x=0..Pi/2, y=0..Pi/2} log(4*cos(x)^2 + 4*cos(y)^2) dy dx = G*Pi, where G is Catalan's constant A006752.

Examples

			6.485696465218497693708581372103315764152266325617976316831738842452555238784...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A071763, A097469, A340182, A340350, A340421, A340422.

Programs

Maple

evalf(Integrate(log(4*cos(x)^2 + 4*cos(y)^2 + 4*cos(z)^2), x = 0..Pi/2, y = 0..Pi/2, z = 0..Pi/2));

PARI

intnum(x = 0, Pi/2, intnum(y = 0, Pi/2, intnum(z = 0, Pi/2, log(4*cos(x)^2 + 4*cos(y)^2 + 4*cos(z)^2)))) \\ 20 valid digits

Formula

Equals limit_{n->infinity} Pi^3 * log(A340182(n)) / (8*n^3).

Equals Pi^3 * log(2)/8 + Integral_{x=0..Pi/2, y=0..Pi/2, z=0..Pi/2} log(3 + cos(2*x) + cos(2*y) + cos(2*z)) dz dy dx.