A002161 -id:A002161 - cp's OEIS frontend

A190732 Decimal expansion of 2/sqrt(Pi).

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

Offset: 1

Alonso del Arte, May 17 2011

According to Weisstein, some mathematicians define erf(z) without reference to this constant.
Also equals the average absolute value of the difference of two independent normally distributed random numbers with mean 0 and variance 1. - Jean-François Alcover, Oct 31 2014
Limit_{n->oo} 2^(1-2*n^2)*n*binomial(2*n^2, n^2) is proper to compute this constant (and also Pi) in a base of power 2. - Ralf Steiner, Apr 23 2017
A gauge point marked "c" on slide rule calculating devices in the 20th century. The Pickworth reference notes its use "in calculating the contents of cylinders". - Peter Munn, Aug 14 2020

			1.12837916709551257...

Chi Keung Cheung et al., Getting Started with Mathematica, 2nd Ed. New York: J. Wiley (2005) p. 79.
C. N. Pickworth, The Slide Rule, 24th Ed., Pitman, London (1945), p 53, Gauge Points.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Steven R. Finch, Mean width of a regular simplex, arxiv:1111.4976 [math.MG], 2016, mu_2.
International Slide Rule Museum, Slide Rule Terms, Glossary and Encyclopedia, entry for "C".
Eric Weisstein's World of Mathematics, Erf
Index entries for transcendental numbers

Cf. A002161, A019704, A285388, A285389.

RealDigits[2/Sqrt[Pi], 10, 100][[1]]
RealDigits[Limit[2^(1 - 2 m^2) m Binomial[2 m^2, m^2], m -> Infinity], 10, 100][[1]] (* Ralf Steiner, Apr 22 2017 *)

2/sqrt(Pi) \\ G. C. Greubel, Jan 09 2017

Equals Sum_{n>=0} (-1)^n*Gamma((n+1)/2)/Gamma(n/2+1). - Jean-François Alcover, Jun 12 2013
Equals 1/A019704. - Michel Marcus, Jan 09 2017
Equals Limit_{n->infinity} A285388(n)/A285389(n). - Ralf Steiner, Apr 22 2017

A271919 Numerator of Product_{j=1..n-1} ((3j+1)/(3j+2)).

Original entry on oeis.org

1, 4, 7, 7, 13, 104, 494, 988, 190, 5320, 20615, 589, 1147, 11470, 246605, 246605, 2416729, 62834954, 4488211, 4488211, 8831641, 10869712, 182067676, 2548947464, 2514502228, 27300309904, 134795280151, 269590560302, 3134773957, 25078191656, 570528860174, 60055669492, 59442856538

Offset: 1

N. J. A. Sloane, May 04 2016

			1, 4/5, 7/10, 7/11, 13/22, 104/187, 494/935, 988/1955, 190/391, 5320/11339,  20615/45356, 589/1334, 1147/2668, 11470/27347, ...

Jan de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.

Sequences of fractions from de Gier paper: A271919-A271926.

Cf. A271920 (denominators), A002161, A203145.

f:=proc(n) local j;
mul(((3*j+1)/(3*j+2)),j=1..n-1); end;
t1:=[seq(f(n),n=1..50)];
map(numer,t1);
map(denom,t1);

a[n_] := Product[(3j + 1)/(3j + 2), {j, 1, n - 1}] // Numerator;
Array[a, 33] (* Jean-François Alcover, Nov 17 2017 *)

a(n) = numerator(prod(j=1, n-1, ((3*j+1)/(3*j+2)))); \\ Michel Marcus, Nov 17 2017

a(n)/A271920(n) ~ c * (4/n)^(1/3), where c = Gamma(5/6)/sqrt(Pi) = A203145/A002161. - Amiram Eldar, Aug 17 2025

A019707 Decimal expansion of sqrt(Pi)/5.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

With offset 1 this is the decimal expansion of 2*sqrt(Pi) = 3.544907..., which is the smallest possible perimeter index eta=P/sqrt(A) of all figures (not necessarily connected) in the Euclidean plane with a continuous boundary of length P (perimeter) enclosing a finite area A. The smallest value is attained only by a Euclidean planar disk. For example, eta=4 for squares, eta=2(sqrt(a/b)+sqrt(b/a))>=4 for aXb rectangles, and eta=4.559014... (A268604) for equilateral triangles. - Stanislav Sykora, Feb 08 2016

			0.3544907701811...= 0.2*A002161.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. A000796, A002161, A087198, A268604.

RealDigits[Sqrt[Pi]/5, 10, 100][[1]] (* Alonso del Arte, Jun 10 2012 *)

sqrt(Pi)/5 \\ Charles R Greathouse IV, Sep 28 2022

sqrt(Pi)/5 = sqrt(4 * Pi)/10.

Equals -Gamma(-1/2)/10, where Gamma is Euler's gamma function. - Lee A. Newberg, Mar 05 2024

A092039 Decimal expansion of cube root of Pi.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 27 2004

This is the diameter of a sphere with volume = (Pi^2)/6 = zeta(2) = A013661. - Eric Desbiaux, Jan 21 2009

Edge of a cube with volume Pi. - Omar E. Pol, Aug 09 2012

			1.4645918875615232630...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A000796 (Pi), A091925 (Pi^3), A093204 (Pi^(-1/3)), A002161 (sqrt(Pi)), A197111 (cont.frac.).

Maple
```
evalf(root[3](Pi)) ;
```

RealDigits[N[Pi^(1/3),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)

sqrtn(Pi,3) \\ Charles R Greathouse IV, Apr 14 2014

1/A093204. - M. F. Hasler, Oct 07 2014

A175574 Decimal expansion of sqrt(Pi) / (Gamma(3/4))^2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jul 15 2010

Entry 34 c of chapter 11 of Ramanujan's second notebook.

This constant is also the ratio T(Pi/2)/T(0), where T(Pi/2) is the exact pendulum period for an amplitude of Pi/2 and T(0) the approximate period 2*Pi*sqrt(L/g) for small angles. - Jean-François Alcover, Aug 05 2014

			1.18034059901609622604533794..

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.
Claudio Carvalhaes and Patrick Suppes, Approximations for the period of the simple pendulum based on the arithmetic-geometric mean, American Journal of Physics 76 (2008), 1150-1154.

Cf. A002161, A068465, A085565, A096427, A175573.

sqrt(pi)/gamma(3/4)^2 % Altug Alkan, Dec 05 2015

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

First@ RealDigits[N[Sqrt@ Pi/Gamma[3/4]^2, 120]] (* Michael De Vlieger, Dec 06 2015 *)

sqrt(Pi)/gamma(3/4)^2 \\ Altug Alkan, Dec 05 2015

Equals A002161 /A068465^2.
Equals 2F1([1/2,1/2],[1],1/2) = 1/agm(1, sqrt(1/2)) = gamma(1/4)^2/(2*Pi^(3/2)).
Equals 2*sqrt(2)*K(-1)/Pi, where K is the complete elliptic integral of the first kind, K(-1) being A085565. - Jean-François Alcover, Jun 03 2014
Equals Product_{k>=1} (1-(-1)^k/(2*k)) =  3/2 * 3/4 * 7/6 * 7/8 * 11/10 * 11/12 * ... .  - Richard R. Forberg, Dec 05 2015
Reciprocal of A096427. Equals ( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2, a rapidly converging series. For example, summing from n = -5 to n = 5 gives the constant correct to 49 decimal places. - Peter Bala, Mar 06 2019
Equals Sum_{k>=0} binomial(2*k,k)^2/2^(5*k). - Amiram Eldar, Aug 26 2020
Equals (3/2)*hypergeom([-1/4, 3/4], [3/2], 1). - Peter Bala, Mar 04 2022
Equals A175573^2. - Amiram Eldar, Jul 04 2023

A-number typo for sqrt(Pi) corrected by R. J. Mathar, Aug 01 2010

A269545 Decimal expansion of Gamma(Pi).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Feb 29 2016

			2.2880377953400324179595889090602339228896881533562224...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A269546, A269547, A269557, A269558, A269559, A257955, A257957, A257958, A257959, A002161, A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; gamma(pi)
```
Maple
```
evalf(GAMMA(Pi), 120);
```
Mathematica
```
RealDigits[Gamma[Pi], 10, 120][[1]]
```
PARI
```
default(realprecision, 120); gamma(Pi)
```

Equals Integral_{x >= 0} x^(Pi-1)/e^x dx (Euler integral of the second kind).

A092040 Decimal expansion of Pi^(1/4).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 27 2004

The convergents of the continued fractions are 4/3, 221/166, 667/501, 888/667, 1555/1168, 3998/3003, 13549/10177, 505311/379552, ... - R. J. Mathar, Jan 29 2025

			1.3313353638003897127975349179502808533093662238181042...

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
John H. Conway and Michael J. T. Guy, Pi in Four 4's, Eureka, The Archimedeans' Journal, Vol. 25 (1962), pp. 18-19.
Index entries for transcendental numbers

Cf. A000796, A092425, A002161.

RealDigits[(Pi)^(1/4), 10, 100][[1]] (* Vincenzo Librandi, Mar 01 2013 *)

Equals Gamma(1/2)^(1/2). - Bruno Berselli, Mar 01 2013

Equals lim_{n->oo} 2^n*n!/(sqrt((2*n)!)*n^(1/4)) (Conway and Guy, 1962). - Amiram Eldar, Feb 19 2022

A098325 Recurrence sequence based on positions of digits in decimal places of sqrt(Pi).

Original entry on oeis.org

0, 9, 10, 75, 39, 218, 78, 61, 45, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 03 2004

			sqrt(Pi)=1.7724538509055...
So for example, a(2)=9 because 9th decimal place of sqrt(Pi) is 0.
a(3)=10 because 10th decimal place of sqrt(Pi) is 9, a(4)=75 because 10 appears at the 75th to 76th decimal places and so on.
This sequence, like the one for Zeta(3) (A098290), repeats after just a few terms once the sequence hits 4 at position 4.

Other recurrence sequences: A097614 for Pi, A098266 for e, A098289 for log(2), A098290 for Zeta(3), A098319 for 1/Pi, A098320 for 1/e, A098321 for gamma, A098322 for G, A098323 for 1/G, A098324 for Golden Ratio, phi. A002161 for digits of sqrt(Pi).

with(StringTools): Digits:=1000: G:=convert(evalf(sqrt(Pi)),string): a[0]:=0: for n from 1 to 15 do a[n]:=Search(convert(a[n-1],string), G)-2:printf("%d, ",a[n-1]):od: # Nathaniel Johnston, Apr 30 2011

a(1)=0, p(i)=position of first occurrence of a(i) in decimal places of sqrt(Pi), a(i+1)=p(i).

A269546 Decimal expansion of log(Gamma(Pi)).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Feb 29 2016

Gamma(x) is the Gamma function (Euler's integral of the second kind).

			0.8276945923234371015295785584523599511535017341207373...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A269545, A269547, A269557, A269558, A269559, A257955, A257957, A257958, A257959, A002161.

Cf. A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; log(gamma(pi))
```
Maple
```
evalf(lnGAMMA(Pi), 120);
```
Mathematica
```
RealDigits[LogGamma[Pi], 10, 120][[1]]
```

default(realprecision, 120); lngamma(Pi)

A269547 Decimal expansion of Psi(Pi).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Feb 29 2016

Psi(x) is the digamma function (logarithmic derivative of the Gamma function).

			0.9772133079420067332920694864061823436408346099943256...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A269545, A269546, A269557, A269558, A269559, A257955, A257957, A257958, A257959, A002161.

Cf. A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; psi(pi)
```
Maple
```
evalf(Psi(Pi), 120)
```
Mathematica
```
RealDigits[PolyGamma[Pi], 10, 120][[1]]
```
PARI
```
default(realprecision, 120); psi(Pi)
```

cp's OEIS Frontend

A190732 Decimal expansion of 2/sqrt(Pi).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A271919 Numerator of Product_{j=1..n-1} ((3*j+1)/(3*j+2)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A019707 Decimal expansion of sqrt(Pi)/5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A092039 Decimal expansion of cube root of Pi.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A175574 Decimal expansion of sqrt(Pi) / (Gamma(3/4))^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

Formula

Extensions

A269545 Decimal expansion of Gamma(Pi).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

Formula

A092040 Decimal expansion of Pi^(1/4).

A271919 Numerator of Product_{j=1..n-1} ((3j+1)/(3j+2)).