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

%I A363747 #54 Mar 27 2025 05:44:04
%S A363747 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,
%T A363747 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,
%U A363747 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
%N A363747 Decimal expansion of 2*Integral_{x=0..1} 1/sqrt(1-x^16) dx.
%C A363747 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.
%C A363747 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
%F A363747 Equals Beta(1/16,1/2)/8 = 2*sqrt(Pi)*Gamma(17/16)/Gamma(9/16).
%F A363747 Equals Pi^(3/2)/(8*sin(Pi/16)*Gamma(9/16)*Gamma(15/16)). - _Christian N. Hofmann_, Aug 28 2023
%F A363747 Gamma(1/2^n) = 2^((n-1)*(1-1/2^n)) * Product_{k=1..n} I(k)^(1/(2^(n-k+1))).
%e A363747 2.1682048381784119930017239089489332786586588673422...
%e A363747 Gamma(1/16) = 2^(45/16)*Pi^(1/16)*I(2)^(1/8)*I(3)^(1/4)*I(4)^(1/2) = 15.481281...
%p A363747 evalf(2*int(1/sqrt(1-t^16),t=0..1), 120);
%t A363747 RealDigits[Beta[1/16, 1/2]/8, 10, 120][[1]] (* _Amiram Eldar_, Jun 22 2023 *)
%o A363747 (PARI) 2*intnum(x=0, 1, 1/sqrt(1-x^16)) \\ _Michel Marcus_, Jun 22 2023
%Y A363747 Cf. A000796, A062539, A262427.
%K A363747 nonn,cons
%O A363747 1,1
%A A363747 _Christian N. Hofmann_, Jun 19 2023