A002161 -id:A002161 - cp's OEIS frontend

A269557 Decimal expansion of Gamma(log(2)).

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

Offset: 1

Iaroslav V. Blagouchine, Feb 29 2016

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

			1.3090409112814812698245325213959295756125890319181890...

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

Cf. A269558, A269559, A269545, A269546, A269547, A257955, A257957, A257958, A257959, A002161.

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

MATLAB
```
format long; gamma(log(2))
```
Maple
```
evalf(GAMMA(ln(2)), 120);
```
Mathematica
```
RealDigits[Gamma[Log[2]], 10, 120][[1]]
```

default(realprecision, 120); gamma(log(2))

A269558 Decimal expansion of log(Gamma(log(2))).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Feb 29 2016

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

			0.2692947402831312429499165832117128248889035102111661...

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

Cf. A269557, A269559, A269545, A269546, A269547, A257955, A257957, A257958, A257959, A002161.

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

MATLAB
```
format long; log(gamma(log(2)))
```
Maple
```
evalf(lnGAMMA(ln(2)), 120);
```

RealDigits[LogGamma[Log[2]], 10, 120][[1]]

default(realprecision, 120); lngamma(log(2))

A269559 Decimal expansion of Psi(log(2)), negated.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Feb 29 2016

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

			-1.2395972796176185082441275516860842454332895226874208...

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

Cf. A269557, A269558, A269545, A269546, A269547, A257955, A257957, A257958, A257959, A002161.

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

MATLAB
```
format long; psi(log(2))
```
Maple
```
evalf(Psi(ln(2)), 120);
```

RealDigits[PolyGamma[Log[2]], 10, 120][[1]]

default(realprecision, 120); psi(log(2))

A037086 Beatty sequence for sqrt(Pi).

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 21, 23, 24, 26, 28, 30, 31, 33, 35, 37, 38, 40, 42, 44, 46, 47, 49, 51, 53, 54, 56, 58, 60, 62, 63, 65, 67, 69, 70, 72, 74, 76, 77, 79, 81, 83, 85, 86, 88, 90, 92, 93, 95, 97, 99, 101, 102, 104, 106, 108, 109, 111, 113, 115, 116

Offset: 0

Felice Russo

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for sequences related to Beatty sequences

Cf. A002161.

A037086:=n->floor(n*sqrt(Pi)): seq(A037086(n), n=0..100); # Wesley Ivan Hurt, Nov 20 2014

Table[Floor[n*Sqrt[Pi]], {n, 0, 100}] (* Wesley Ivan Hurt, Nov 20 2014 *)

a(n) = floor(n*sqrt(Pi)); \\ Michel Marcus, Sep 28 2013

a(n) = floor(n*sqrt(Pi)).

More terms from James Sellers, Jul 06 2000

A175476 Decimal expansion of Pi^(3/2).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, May 25 2010

Log(Pi^(3/2)) = 1.5*log(Pi) = 1.5 * A053510 = 1.7170948...

			5.5683279968317078452848179..

Marc Chamberland and Armin Straub, On gamma quotients and infinite products, Advances in Applied Mathematics, Vol. 51, No. 5 (2013), pp. 546-562, see pp. 555-556; arXiv preprint, arXiv:1309.3455 [math.NT], 2013, see pp. 9-10.
Index entries for transcendental numbers.

Cf. A000796, A002161, A002388, A053510.

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

RealDigits[Pi^(3/2), 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

Equals A000796 * A002161 = A002388 / A002161.
From Vaclav Kotesovec, Dec 10 2015: (Start)
Equals Gamma(3/14)*Gamma(5/14)*Gamma(13/14)/2.
Equals Gamma(1/14)*Gamma(9/14)*Gamma(11/14)/4.
(End)
Equals Integral_{x=-oo..oo, y=-oo..oo, z=-oo..oo} exp(-x^2 - y^2 - z^2) dx dy dz. - Ilya Gutkovskiy, Apr 10 2024

A195907 Decimal expansion of Sum_{n = -oo..oo} exp(-n^2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 25 2011

A Riemann sum approximation to Integral_{-oo..oo} exp(-x^2) dx = sqrt(Pi).

			1.77263720482665215303125055115785848134338604537224605383159051...
For comparison, sqrt(Pi) = 1.7724538509055160272981674833411451827975494561223871282138... (A002161).

Mentioned by N. D. Elkies in a lecture on the Poisson summation formula in Nashville TN in May 2010.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Dedekind Eta Function
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A002161, A218792.

N[Sum[Exp[-n^2], {n, -Infinity, Infinity}], 200]
RealDigits[ N[ EllipticTheta[3, 0, 1/E], 115]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

1 + 2*suminf(n=1,exp(-n^2)) \\ Charles R Greathouse IV, Jun 06 2016

(eta(I/Pi))^5 / (eta(I/(2*Pi))^2 * eta(2*I/Pi)^2) \\ Jianing Song, Oct 13 2021

Equals Jacobi theta_{3}(0,exp(-1)). - Jianing Song, Oct 13 2021
Equals eta(i/Pi)^5 / (eta(i/(2*Pi))*eta(2*i/Pi))^2, where eta(t) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + ... is the Dedekind eta function without the q^(1/24) factor in powers of q = exp(2*Pi*i*t) (Cf. A000122). - Jianing Song, Oct 14 2021
Equals Product_{k>=1} tanh((k*(1 + i*Pi))/2), i=sqrt(-1). - Antonio Graciá Llorente, May 13 2024

A217481 Decimal expansion of sqrt(2*Pi)/4.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 04 2012

Equals Integral_{x>=0} sin(x^2) dx.
The generalizations are Integral_{x>=0} exp(i*x^n) dx =
  0.6266570686577501... + i*0.6266570686577501... for n=2,
  0.7733429420779898... + i*0.4464897557846246... for n=3,
  0.8374066967690864... + i*0.3468652110238094... for n=4,
  0.8732303655178185... + i*0.2837297451053993... for n=5,
and
  Gamma(1/n)*i^(1/n)/n in general, where i is the imaginary unit. - R. J. Mathar, Nov 14 2012
Mean of cycle length (and of tail length) in Pollard rho method for factoring n is sqrt(2*Pi)/4*sqrt(n). - Jean-François Alcover, May 27 2013
If m = (1/2) * sqrt(Pi/2), then the coordinates of the two asymptotic points of the Cornu spiral (also called clothoide) and whose Cartesian parametrization is: x = a * Integral_{0..t} cos(u^2) du and y = a * Integral_{0..t} sin(u^2) du are (a*m, a*m) and (-a*m, -a*m) (see the curve at the MathCurve link). - Bernard Schott, Mar 02 2020
Equals the limit as x approaches infinity of the Fresnel integrals Integral_{0..x} sin(t^2) dt and Integral_{0..x} cos(t^2) dt. - Bernard Schott, Mar 05 2020

			equals 0.62665706865775012560394132120276131... = A019727 / 4 = sqrt(A019675).

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Robert Ferréol, Cornu spiral, Mathcurve.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 420 (formulas 3.757.1, 3.757.2).
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.4.7)
Wikipedia, Fresnel Integral
Index entries for transcendental numbers

Apart from possible scaling sqrt(A019692/2^n) for n=0..7 are A019727, A002161, A069998, A019704, this sequence, A019706, A143149, A019710.

Sqrt(2*Pi(RealField(100)))/4; // G. C. Greubel, Sep 30 2018

Maple
```
evalf(sqrt(2*Pi))/4 ;
```

First@ RealDigits[N[Sqrt[2 Pi]/4, 105]] (* Michael De Vlieger, Sep 24 2018 *)

fpprec : 100; ev(bfloat(sqrt(2*%pi)))/4; /* Martin Ettl, Oct 04 2012 */

sqrt(2*Pi)/4 \\ Altug Alkan, Sep 08 2018

((sqrt(2*pi))/4).n(digits=100) # Jani Melik, Oct 05 2012

From A.H.M. Smeets, Sep 22 2018: (Start)
Equals Integral_{x >= 0} sin(4x)/sqrt(x) dx [see Gradsteyn and Ryzhik].
Equals Integral_{x >= 0} cos(4x)/sqrt(x) dx [see Gradsteyn and Ryzhik]. (End)
From Bernard Schott, Mar 02 2020: (Start)
Equals Integral_{x >= 0} cos(x^2) dx or Integral_{x >= 0} sin(x^2) dx.
Equals sqrt(Pi/8) or (1/2)*sqrt(Pi/2). (End)

A249521 Decimal expansion of 4/sqrt(Pi), the average distance between two random Gaussian points in three dimensions.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 31 2014

Coordinates are independent normally distributed random variables with mean 0 and variance 1.

			2.25675833419102514779231780624309034337620251731...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Steven R. Finch, Random Triangles, Jan 21 2010. [Cached copy, with permission of the author]
Index entries for transcendental numbers.

Cf. A002161 (the analog constant in two dimensions), A087197, A190732 (the analog constant in one dimension).

RealDigits[4/Sqrt[Pi], 10, 103] // First

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

Equals Sum_{k>=0} k!/(k+3/2)!. - Amiram Eldar, Jun 19 2023

A019710 Decimal expansion of sqrt(Pi)/8.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

Cf. A002161.

RealDigits[Sqrt[Pi]/8,10,120][[1]] (* Harvey P. Dale, Jul 14 2011 *)

sqrt(Pi)/8 \\ Charles R Greathouse IV, Sep 30 2022

Equals 10 * Sum_{k>=1} (k+1/2)!/(k+4)!. - Amiram Eldar, Jun 19 2023

A068450 Factorial expansion of sqrt(Pi) = Sum_{n>0} a(n)/n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 1, 1, 3, 0, 5, 10, 6, 8, 12, 0, 10, 0, 12, 9, 6, 12, 22, 21, 24, 3, 14, 21, 13, 24, 21, 11, 8, 22, 27, 3, 8, 1, 36, 21, 27, 15, 2, 41, 22, 34, 8, 0, 4, 8, 39, 48, 27, 38, 22, 0, 23, 49, 19, 27, 29, 28, 40, 33, 21, 62, 7, 67, 33, 8, 30, 18, 60, 73, 61, 72, 42, 59, 22

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

			sqrt(Pi) = 1 + 1/2! + 1/3! + 2/4! + 2/5! + 4/6! + 1/7! + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A075874, A002161 (decimal expansion).

SetDefaultRealField(RealField(250));  R:= RealField(); [Floor(Sqrt(Pi(R)))] cat [Floor(Factorial(n)*Sqrt(Pi(R))) - n*Floor(Factorial((n-1))*Sqrt(Pi(R))) : n in [2..30]]; // G. C. Greubel, Mar 21 2018

Table[If[n == 1, Floor[Sqrt[Pi]], Floor[n!*Sqrt[Pi]] - n*Floor[(n - 1)!*Sqrt[Pi]]], {n, 1, 50}] (* G. C. Greubel, Mar 21 2018 *)

default(realprecision, 250); for(n=1,30, print1(if(n==1, floor(sqrt(Pi)), floor(n!*sqrt(Pi)) - n*floor((n-1)!*sqrt(Pi))), ", ")) \\ G. C. Greubel, Mar 21 2018

vector(30,n,if(n>1,t=t%1*n,t=sqrt(Pi))\1) \\ M. F. Hasler, Nov 25 2018

def A068450(n):
    if (n==1): return floor(sqrt(pi))
    else: return expand(floor(factorial(n)*sqrt(pi)) - n*floor(factorial(n-1)*sqrt(pi)))
[A068450(n) for n in (1..80)] # G. C. Greubel, Nov 27 2018

Keyword cons removed by R. J. Mathar, Jul 23 2009

cp's OEIS Frontend

A269557 Decimal expansion of Gamma(log(2)).

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A269558 Decimal expansion of log(Gamma(log(2))).

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A269559 Decimal expansion of Psi(log(2)), negated.

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A037086 Beatty sequence for sqrt(Pi).

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A175476 Decimal expansion of Pi^(3/2).

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A195907 Decimal expansion of Sum_{n = -oo..oo} exp(-n^2).

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A217481 Decimal expansion of sqrt(2*Pi)/4.

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