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A363747
Decimal expansion of 2*Integral_{x=0..1} 1/sqrt(1-x^16) dx.
0
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
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
KEYWORD
nonn,cons
AUTHOR
STATUS
approved