A363747 Decimal expansion of 2*Integral_{x=0..1} 1/sqrt(1-x^16) dx.

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

Offset

1,1

Comments

Let I(k) = 2*Integral_{x=0..1} 1/sqrt(1-x^(2^k)) dx. Then I(1) = Pi (cf. A000796), I(2) = Gauss lemniscate constant (cf. A062539), I(3) = sqrt(2)*K(sqrt(2)-1) (cf. A262427), I(4) = this constant.

Also I(k) = (Pi^(3/2)*Product_{j=2..k} cos(Pi/(2^j)))/(Gamma(1/2+1/(2^k))*Gamma(1-1/(2^k)). - Christian N. Hofmann, Aug 25 2023

Links

Table of n, a(n) for n=1..105.

Formula

Equals Beta(1/16,1/2)/8 = 2*sqrt(Pi)*Gamma(17/16)/Gamma(9/16).

Equals Pi^(3/2)/(8*sin(Pi/16)*Gamma(9/16)*Gamma(15/16)). - Christian N. Hofmann, Aug 28 2023

Gamma(1/2^n) = 2^((n-1)*(1-1/2^n)) * Product_{k=1..n} I(k)^(1/(2^(n-k+1))).

Example

2.1682048381784119930017239089489332786586588673422...
Gamma(1/16) = 2^(45/16)*Pi^(1/16)*I(2)^(1/8)*I(3)^(1/4)*I(4)^(1/2) = 15.481281...

Maple

evalf(2*int(1/sqrt(1-t^16), t=0..1), 120);

Mathematica

RealDigits[Beta[1/16, 1/2]/8, 10, 120][[1]] (* Amiram Eldar, Jun 22 2023 *)

Prog

(PARI) 2*intnum(x=0, 1, 1/sqrt(1-x^16)) \\ Michel Marcus, Jun 22 2023

Crossrefs

Cf. A000796, A062539, A262427.

Sequence in context: A112007 A113374 A136470 * A220884 A217877 A138510

Adjacent sequences: A363744 A363745 A363746 * A363748 A363749 A363750

Keyword

nonn,cons

Author

Christian N. Hofmann, Jun 19 2023

Status

approved