A186642 Decimal expansion of the "squircle" perimeter.

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, 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, 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

Offset

1,1

Comments

This squircle constant can also be computed as a series in terms of incomplete beta function with coefficients from sequences A002596 and A120777:

a(n) = (-1)^(n+1) numerator((2n-3)!!/n!) ( sequence A002596);

b(n) = denominator(binomial(2n+2, n+1)/2^(2n+1)) ( sequence A120777).

Generic term:

u(n) = (a(n)/b(n-1))*beta(1/2, (6n+1)/4, 1-(3/2)*n).

Here is the series computed up to 5 terms:

4*2^(3/4) + sum(u(n), {n, 1, 5}) =

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).

It evaluates to 7.018901897260651...

Numeric check with 10000 terms:

4*2^(3/4) + sum(u(n), {n, 1, 10000}) = 7.017697943556135...

Links

Table of n, a(n) for n=1..100.

Eric Weisstein's World of Mathematics, Squircle

Formula

-((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

Example

7.01769794356404...

Mathematica

First @ RealDigits[N[2*Integrate[Sqrt[1 + x^(3/2)/(1 - x)^(3/2)]/x^(3/4), {x, 0, 1/2}], 100]]
(* This other series formula gives 100 correct digits: *)
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]

Crossrefs

Cf. A175576 (unit squircle area).

Sequence in context: A202354 A324498 A243908 * A153626 A297787 A101031

Adjacent sequences: A186639 A186640 A186641 * A186643 A186644 A186645

Keyword

cons,easy,nonn

Author

Jean-François Alcover, Feb 25 2011

Status

approved