A175476 -id:A175476 - cp's OEIS frontend

A175576 Decimal expansion Pi^(3/2)/Gamma(3/4)^2.

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

Offset: 1

R. J. Mathar, Jul 15 2010

Entry 34 e of chapter 11 of Ramanujan's second notebook.
In addition, Pi^(3/2) / Gamma(3/4)^2 is the area of the unit "squircle" as defined in MathWorld. (Note that 8*Gamma(5/4)^2 / sqrt(Pi) is the same constant.) - Jean-François Alcover, Feb 24 2011
Real period of the elliptic curve y^2 = x*(x - 1)*(x - 1/2). See Rouse. - Peter Bala, Dec 06 2024

			3.708149354602743836867700694390520092435197647...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Bruce C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. Lond. Math. Soc. vol 15 no 4 (1983) 273-320.
Jeremy Rouse, Hypergeometric functions and elliptic curves, Ramanujan Journal, Vol. 12 (2006), pp. 197-205.
Eric Weisstein's World of Mathematics, Squircle

Cf. A062539, A068465, A175476, A175575, A290570.

Maple
```
Pi^(3/2)/GAMMA(3/4)^2 ; evalf(%) ;
```

RealDigits[Pi*EllipticTheta[3, 0, Exp[-Pi]]^2, 10, 50][[1]]
RealDigits[Pi^(3/2)/(Gamma[3/4])^2, 10, 50][[1]] (* G. C. Greubel, Feb 12 2017 *)

Pi^1.5/gamma(3/4)^2 \\ Charles R Greathouse IV, Jun 06 2016

Equals A175476 / A068465^2 = 1/A175575.
Equals Integral_{-oo, oo} 1/(1+2*x^2)^(3/4) or Integral_{-oo, oo} 1/sqrt(1+x^4). - Jean-François Alcover, Jun 04 2013
Equals sqrt(2)*L, where L is the lemniscate constant A062539. - Jean-François Alcover, Aug 11 2014
From Peter Bala, Mar 01 2022 : (Start)
Equals 3*Sum_{n >= 0} (1/(4*n+1) + 1/(4*n-3))*binomial(1/2,n). Cf. A290570.
Equals hypergeom([-1/2, 3/4, -3/4], [-1/4, 5/4], -1).
Equals 2*hypergeom([1/4, 3/4], [5/4], 1) = (16/5)*hypergeom([-1/4, -3/4], [5/4], 1). (End)
Equals 2 * A093341. - R. J. Mathar, Dec 08 2023
From Peter Bala, Dec 06 2024: (Start)
Equals Pi*hypergeom([1/2, 1/2], [1], 1/2).
Equals 2*Integral_{x = 0..Pi/2} 1/sqrt(1 - (1/2)*sin^2(x)) dx. See Rouse. (End)

A220610 Decimal expansion of sqrt(2*Pi^3).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Dec 25 2012

This is the case n=4 of Product_{i=1..n-1} Gamma(i/n) = sqrt((2*Pi)^(n-1)/n).

Continued fraction expansion: 7, 1, 6, 1, 79, 4, 7, 1, 1, 1, 1, 1, 1, 4, 2, 3, 73, 1, 2, 1, 14, 3, 2, 1, 1, 2, 3, 1, ...

			7.8748049728612098721453229972336022711558426913993669291...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wikipedia, Particular values of the Gamma function: Products
Index entries for transcendental numbers

Cf. numbers of the form sqrt((2*Pi)^(n-1)/n) -- see the first comment: A002161 (n=2), A186706 (n=3).

SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(2*Pi(R)^3); // G. C. Greubel, Sep 29 2018

evalf(sqrt(2*Pi^3),120); # Muniru A Asiru, Sep 30 2018

Mathematica
```
RealDigits[Sqrt[2 Pi^3], 10, 90][[1]]
```
Maxima
```
fpprec:90; ev(bfloat(sqrt(2*%pi^3)));
```

default(realprecision, 100); sqrt(2*Pi^3) \\ G. C. Greubel, Sep 29 2018

Equals A002193*A175476.

A371534 Decimal expansion of (1/8) * Pi^(3/2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 10 2024

			0.6960409996039634806606022477648544627517030487854...

Index entries for transcendental numbers

Cf. A003881, A019704, A175476, A222068.

RealDigits[(1/8) Pi^(3/2), 10, 103][[1]]

Equals Integral_{x=0..oo, y=0..oo, z=0..oo} exp(-x^2 - y^2 - z^2) dx dy dz.

A385568 Decimal expansion of Pi^(3/4).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jul 03 2025

			2.35973049241469688757847446452193444245804053693781...

Index entries for transcendental numbers

Cf. A092040, A175476, A196535.

Mathematica
```
RealDigits[Pi^(3/4), 10, 105][[1]]
```

Equals sqrt(A175476).

Equals A092040^3.

cp's OEIS Frontend

A175576 Decimal expansion Pi^(3/2)/Gamma(3/4)^2.

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A220610 Decimal expansion of sqrt(2*Pi^3).

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A371534 Decimal expansion of (1/8) * Pi^(3/2).

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A385568 Decimal expansion of Pi^(3/4).

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