cp's OEIS Frontend

A186642 Decimal expansion of the "squircle" perimeter.

%I A186642 #27 Feb 16 2025 08:33:14
%S A186642 7,0,1,7,6,9,7,9,4,3,5,6,4,0,4,1,6,4,7,1,0,6,4,9,4,1,6,3,9,3,1,8,1,1,
%T A186642 6,9,3,9,8,0,0,8,7,5,0,4,9,7,2,4,4,9,3,4,3,2,2,8,8,6,1,0,3,5,6,0,7,3,
%U A186642 9,2,2,1,1,6,1,8,1,8,8,8,3,5,1,3,2,3,8,8,3,9,3,0,0,5,0,3,4,0,7,1
%N A186642 Decimal expansion of the "squircle" perimeter.
%C A186642 This squircle constant can also be computed as a series in terms of incomplete beta function with coefficients from sequences A002596 and A120777:
%C A186642   a(n) = (-1)^(n+1) numerator((2n-3)!!/n!) ( sequence A002596);
%C A186642   b(n) = denominator(binomial(2n+2, n+1)/2^(2n+1)) ( sequence A120777).
%C A186642   Generic term:
%C A186642   u(n) = (a(n)/b(n-1))*beta(1/2, (6n+1)/4, 1-(3/2)*n).
%C A186642   Here is the series computed up to 5 terms:
%C A186642   4*2^(3/4) + sum(u(n), {n, 1, 5}) =
%C A186642   4*2^(3/4) + beta(1/2, 7/4, -1/2) - (1/4)*beta(1/2, 13/4, -2) + (1/8)* beta(1/2, 19/4, -7/2) - (5/64)*beta(1/2, 25/4, -5) + (7/128)*beta(1/2, 31/4, -13/2).
%C A186642   It evaluates to 7.018901897260651...
%C A186642   Numeric check with 10000 terms:
%C A186642   4*2^(3/4) + sum(u(n), {n, 1, 10000}) = 7.017697943556135...
%H A186642 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squircle.html">Squircle</a>
%F A186642 -((3^(1/4) MeijerG[{{1/3, 2/3, 5/6, 1, 4/3}, {}}, {{1/12, 5/12, 7/12, 3/4, 13/12}, {}}, 1])/(16 Sqrt[2] Pi^(7/2) Gamma[5/4])). - _Eric W. Weisstein_, Oct 25 2011
%e A186642 7.01769794356404...
%t A186642 First @ RealDigits[N[2*Integrate[Sqrt[1 + x^(3/2)/(1 - x)^(3/2)]/x^(3/4), {x, 0, 1/2}], 100]]
%t A186642 (* This other series formula gives 100 correct digits: *)
%t A186642 First @ RealDigits[1/Sqrt[Pi]*NSum[(-1)^(n+1)*Gamma[n - 1/2]*Beta[1/2, (6n + 1)/4, 1 - (3/2)n] / n!, {n, 0, Infinity},WorkingPrecision -> 100, Method -> "AlternatingSigns"], 10, 100]
%Y A186642 Cf. A175576 (unit squircle area).
%K A186642 cons,easy,nonn
%O A186642 1,1
%A A186642 _Jean-François Alcover_, Feb 25 2011