A231863 -id:A231863 - cp's OEIS frontend

A019727 Decimal expansion of sqrt(2*Pi).

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

Offset: 1

N. J. A. Sloane

Pickover says that the expression: lim_{n->oo} e^n(n!) / (n^n * sqrt(n)) = sqrt(2*Pi) is beautiful because it connects Pi, e, radicals, factorials and infinite limits. - Jason Earls, Mar 16 2001
Appears in the formula of the normal distribution. - Johannes W. Meijer, Feb 23 2013
sqrt(2*Pi)*sqrt(n) is the expected height of a labeled random tree of order n (see Rényi, Szekeres, 1967, formula (4.6)). - Hugo Pfoertner, May 18 2023
The constant in the formula known as "Stirling's approximation" (or "Stirling's formula"). It is sometimes called Stirling constant. The formula without the exact value of the constant was discovered by the French mathematician Abraham de Moivre (1667-1754), and was published in his book (1730). The exact value of the constant was found by the Scottish mathematician James Stirling (1692-1770) and was published in his book "Methodus differentialis" (1730). - Amiram Eldar, Jul 08 2023

			2.506628274631000502415765284811045253006986740609938316629923576342293....

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
Clifford A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 307.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Mohammad K. Azarian, An Expression for Pi, Problem #870, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. Solution appeared in Vol. 40, No. 1, January 2009, pp. 62-64.
Abraham de Moivre, Miscellanea Analytica de Seriebus et Quadraturis, London, England: J. Tonson & J. Watts, 1730, pp. 96-106.
K. Kimoto, N. Kurokawa, C. Sonoki, and M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.
Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
A. Rényi and G. Szekeres, On the height of trees, Journal of the Australian Mathematical Society , Volume 7 , Issue 4 , November 1967 , pp. 497-507.
James Stirling, Methodus differentialis, sive tractatus de summatione et interpolatione serierum infinitarum, London, 1730. See Propositio XXVIII, pp. 135-139.
Eric Weisstein's World of Mathematics, Normal Distribution.
Wikipedia, Stirling's approximation.
Index entries for transcendental numbers.

Cf. A058293 (continued fraction), A231863 (inverse), A000796 (Pi).

R:= RealField(100); Sqrt(2*Pi(R)); // G. C. Greubel, Mar 08 2018

RealDigits[Sqrt[2Pi],10,120][[1]] (* Harvey P. Dale, Dec 12 2012 *)

fpprec: 100$ ev(bfloat(sqrt(2*%pi))); /* Martin Ettl, Oct 11 2012 */

default(realprecision, 20080); x=sqrt(2*Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019727.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009

Equals lim_{n->oo} e^n*(n!)/n^n*sqrt(n).
Also equals Integral_{x >= 0} W(1/x^2) where W is the Lambert function, which is also known as ProductLog. - Jean-François Alcover, May 27 2013
Also equals the generalized Glaisher-Kinkelin constant A_0, see the Finch reference. - Jean-François Alcover, Dec 23 2014
Equals exp(-zeta'(0)). See Kimoto et al. - Michel Marcus, Jun 27 2019

A203142 Decimal expansion of Gamma(1/8).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 29 2011

			7.5339415987976119046992298412151336246104195881490759409831...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Raimundas Vidunas, Expressions for values of the Gamma function, arXiv:math/0403510 [math.CA], 2004.
Wikipedia, Particular values of the Gamma function: General rational arguments
Index to sequences related to the Gamma function

SetDefaultRealField(RealField(100)); Gamma(1/8); // G. C. Greubel, Mar 10 2018

RealDigits[Gamma[1/8], 10, 100][[1]] (* Bruno Berselli, Dec 13 2012 *)
RealDigits[Pi^(1/8) * 2^(17/8) * EllipticK[1/2]^(1/4) * EllipticK[3 - 2*Sqrt[2]]^(1/2), 10, 100][[1]] (* Vaclav Kotesovec, Apr 15 2024 *)

default(realprecision, 100); gamma(1/8) \\ G. C. Greubel, Jan 15 2017

this * A203144 * A231863 /2^(1/4) = A068466. - R. J. Mathar, Jan 15 2021

A203145 Decimal expansion of Gamma(5/6).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 29 2011

			1.1287870299081259612609010902588420133267874416647554517520...

Cf. A002161, A019692, A073005, A073006, A175379.

RealDigits[Gamma[5/6], 10, 100][[1]] (* Bruno Berselli, Dec 18 2012 *)

gamma(5/6) \\ Charles R Greathouse IV, Nov 26 2024

A073005 * this * A231863 * A010768 = A073006. - R. J. Mathar, Jan 15 2021
Equals 2*Pi/Gamma(1/6) = A019692 / A175379. - Amiram Eldar, Jul 04 2023
Equals 2^(4/3) * Pi^(3/2) / (sqrt(3) * Gamma(1/3)^2). - Vaclav Kotesovec, Jul 04 2023

A203146 Decimal expansion of Gamma(7/8).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 29 2011

			1.0896523574228969512523767551028929711478700677675651205137...

Index to sequences related to the Gamma function.

Cf. A010767, A068465, A203143, A231863.

RealDigits[Gamma[7/8], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

gamma(7/8) \\ Charles R Greathouse IV, Mar 25 2014

A068465 = A231863 * A010767 * A203143 * this. - R. J. Mathar, Jan 15 2021

A203125 Decimal expansion of (1/8)! = Gamma(9/8).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 29 2011

			.94174269984970148808740373015189170307630244851863449262289...

Index to sequences related to the Gamma function

Cf. A068467, A202623, A203126, A203142.

RealDigits[(1/8)!,10,120][[1]] (* Harvey P. Dale, May 30 2014 *)

gamma(9/8) \\ Charles R Greathouse IV, Mar 25 2014

Equals A203142/8. - R. J. Mathar, Jan 15 2021
A203144 *this *A231863 *A011006 = A068467. - R. J. Mathar, Jan 15 2021
Equals Integral_{x=0..oo} exp(-x^8) dx. - Ilya Gutkovskiy, Sep 18 2021

A203126 Decimal expansion of (1/6)! = Gamma(7/6).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 29 2011

			.92771933363003920070834948253462101856646651914547557693612...

Index to sequences related to the Gamma function

Cf. A068467, A175379, A202623, A203125.

RealDigits[(1/6)!,10,120][[1]] (* Harvey P. Dale, Aug 10 2013 *)

gamma(7/6) \\ Charles R Greathouse IV, Mar 25 2014

Equals A175379/6. - R. J. Mathar, Jan 15 2021
A073006 * this * A231863 * A329219 = A202623. - R. J. Mathar, Jan 15 2021
Equals Integral_{x=0..oo} exp(-x^6) dx. - Ilya Gutkovskiy, Sep 18 2021

A096616 Decimal expansion of 2/3 + zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jun 30 2004

			0.0840695087...

David H. Bailey, Jonathan M. Borwein, Neil J. Calkin, Roland Girgensohn, D. Russell Luke and Victor H. Moll, Experimental Mathematics in Action, Wellesley, MA: A K Peters, 2007, pp. 18 and 227.
Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, Wellesley, MA: A K Peters, 2004, pp. 15-17.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Jonathan M. Borwein and Scott B. Lindstrom, Meetings with Lambert W and other special functions in optimization and analysis, Pure and Applied Functional Analysis, Vol. 1, No. 3 (2016), pp. 361-396, alternative link.
Donald E. Knuth, Problem 10832, The American Mathematical Monthly, Vol. 107, No. 9 (2000), p. 863, A Stirling Series, solution by Cecil C. Rousseau, ibid., Vol. 108, No. 9 (2001), pp. 877-878.
Allen Stenger, Experimental Math for Math Monthly Problems, The American Mathematical Monthly, Vol. 124, No. 2 (2017), pp. 116-131, alternative link.
Eric Weisstein's World of Mathematics, Knuth's Series.

Cf. A019727, A059750, A231863.

Flatten[{0, RealDigits[2/3 + Zeta[1/2]/Sqrt[2*Pi], 10, 100][[1]]}] (* Vaclav Kotesovec, Aug 16 2015 *)

2/3 + zeta(1/2)/sqrt(2*Pi) \\ Michel Marcus, Aug 15 2015

Equals Sum_{k>=1} (1/sqrt(2*Pi*k) - k^k/(k!*exp(k))). - Amiram Eldar, Oct 13 2020

Equals 2/3 - A134469. - R. J. Mathar, Dec 17 2024

A203127 Decimal expansion of (3/8)! = Gamma(11/8).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 29 2011

			0.88891356915622534074242756406624469120777530125959687041567...

Index to sequences related to the Gamma function.

Cf. A011027, A203130, A203143, A203146, A231863.

RealDigits[Gamma[11/8], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

gamma(11/8) \\ Charles R Greathouse IV, Mar 25 2014

Equals 3*A203143/8. - R. J. Mathar, Jan 15 2021
A203146 * this * A231863 * A011027 = A203130. - R. J. Mathar, Jan 15 2021
Equals Integral_{x=0..oo} exp(-x^(8/3)) dx. - Ilya Gutkovskiy, Apr 10 2024

A235916 Decimal expansion of 3/sqrt(2*Pi).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 16 2014

The radius of the large circle, the a-value in the MathWorld link, of a deltoid (3-cusped hypocycloid) with area 1. Thus, for any r > 0, this particular a*sqrt(r) is the radius of the large circle of a deltoid with area r. The radius of the small circle is a*sqrt(r)/3 = A231863*sqrt(r), because A231863 is the radius of the small circle, the b-value in the MathWorld link, of a deltoid with area 1.

			1.1968268412042980338198381798031456054275758934948039729977774890119737...

Rick L. Shepherd, Table of n, a(n) for n = 1..20000
MathWorld, Deltoid
Index entries for transcendental numbers

Cf. A019727, A019728, A231863 (corresponding small circle radius).

SetDefaultRealField(RealField(100)); R:= RealField(); 3/Sqrt(2*Pi(R)); // G. C. Greubel, Sep 30 2018

Mathematica
```
RealDigits[N[3/Sqrt[2Pi],105]] [[1]]
```

default(realprecision, 120); 3/sqrt(2*Pi)

3/sqrt(2*Pi) = 3/A019727 = 3*A231863 = 1/A019728.

A203132 Decimal expansion of (7/8)! = Gamma(15/8).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 29 2011

			0.95344581274503483234582966071503134975438630929661948044949...

Index to sequences related to the Gamma function

RealDigits[(7/8)!,10,120][[1]] (* Harvey P. Dale, Jun 23 2016 *)

gamma(15/8) \\ Charles R Greathouse IV, Mar 25 2014

Equals 7*A203146/8. - R. J. Mathar, Jan 15 2021
A203127 * this * A231863*2^(9/4) = (7/4)* A203130. - R. J. Mathar, Jan 15 2021
Equals Integral_{x=0..oo} exp(-x^(8/7)) dx. - Ilya Gutkovskiy, Apr 10 2024

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A019727 Decimal expansion of sqrt(2*Pi).

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A203142 Decimal expansion of Gamma(1/8).

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A203145 Decimal expansion of Gamma(5/6).

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A203146 Decimal expansion of Gamma(7/8).

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A203125 Decimal expansion of (1/8)! = Gamma(9/8).

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A203126 Decimal expansion of (1/6)! = Gamma(7/6).

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A096616 Decimal expansion of 2/3 + zeta(1/2)/sqrt(2*Pi).

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