A069998 -id:A069998 - cp's OEIS frontend

A068465 Decimal expansion of Gamma(3/4).

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

Offset: 1

Benoit Cloitre, Mar 10 2002

			Gamma(3/4) = 1.225416702465177645129098303362890526851239248108070611...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:14 at page 414.

G. C. Greubel, Table of n, a(n) for n = 1..20000
Russell J. Matheson, GAMMA(3/4) computed to 14550 digits.
Simon Plouffe, GAMMA(3/4) to 256 places, see p. 65.
Index to sequences related to the Gamma function

Cf. A000796, A002193, A053004, A063448, A068466, A069998.

SetDefaultRealField(RealField(105)); Gamma(3/4); // G. C. Greubel, Mar 11 2018

evalf(GAMMA(3/4)) ; # R. J. Mathar, Jan 10 2013

RealDigits[Gamma[3/4], 10, 100][[1]] (* G. C. Greubel, Mar 11 2018 *)

default(realprecision, 100); gamma(3/4) \\ G. C. Greubel, Mar 11 2018

Gamma(3/4) * A068466 = sqrt(2)*Pi = A063448. - R. J. Mathar, Jun 18 2006
Equals Integral_{x>=0} x^(-1/4)*exp(-x) dx. - Vaclav Kotesovec, Nov 12 2020
Equals (Pi/2)^(1/4) * sqrt(AGM(1,sqrt(2))) = sqrt(A069998 * A053004). - Amiram Eldar, Jun 12 2021

A218792 Decimal expansion of Sum_{n = -oo..oo} e^(-2*n^2).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Nov 05 2012

			1.2713415221890152252223825787909356249768649877176...
For comparison, sqrt(Pi/2) = 1.2533141373155002512078826424055226265034933703050...

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. A195907, A069998, A167009, A167010, A229052.

RealDigits[Sum[E^(-2*k^2), {k,-Infinity,Infinity}], 10, 200][[1]]
RealDigits[EllipticTheta[3,0,1/E^2],10,200][[1]] (* Vaclav Kotesovec, Sep 22 2013 *)

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

(eta(2*I/Pi))^5 / (eta(I/Pi)^2 * eta(4*I/Pi)^2) \\ Jianing Song, Oct 13 2021

Equals Jacobi theta_{3}(0,exp(-2)). - G. C. Greubel, Feb 01 2017

Equals eta(2*i/Pi)^5 / (eta(i/Pi)*eta(4*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

A244067 Decimal expansion of the Purdom-Williams constant, a constant related to the Golomb-Dickman constant and to the asymptotic evaluation of the expectation of a random function longest cycle length.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2014

			0.78248160099165661501621518806291...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 288.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Paul W. Purdom and John H. Williams, Cycle length in a random function, Transactions of the American Mathematical Society, Vol. 133, No. 2 (1968), pp. 547-551.
Eric Weisstein's MathWorld, Golomb-Dickman Constant.
Wikipedia, Golomb-Dickman constant.

Cf. A069998, A084945.

lambda = Integrate[Exp[LogIntegral[x]], {x, 0, 1}]; N[lambda*Sqrt[Pi/2], 99] // RealDigits // First

Equals sqrt(Pi/2)*Integral_{x=0..1} exp(li(x)) dx, where li is the logarithmic integral function.

Equals A069998 * A084945. - Amiram Eldar, Aug 25 2020

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)

A256358 Decimal expansion of log(sqrt(Pi/2)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 26 2015

Equals the derivative of the Dirichlet eta function at x=0. - Stanislav Sykora, May 27 2015

			0.22579135264472743236309761494744107178589733927752815869647153...

G. C. Greubel, Table of n, a(n) for n = 0..5000
J.-F. Alcover, Plot of harmonic sum G(x) for x >= 0 .
Henri Cohen, Fernando Rodriguez Villegas, Don Zagier, Convergence Acceleration of Alternating Series, Exp. Math. 9 (1) (2000) 3-12.
Costas Efthimiou, Problem 1838, Mathematics Magazine, Vol. 83, No. 1 (2010), p. 65; A weakly convergent series of logs, Solution to Problem 1838 by Tiberiu Trif, ibid., Vol. 84, No. 1 (2011), pp. 65-67.
Philippe Flajolet, Xavier Gourdon, and Philippe Dumas, Mellin Transforms and Asymptotics: Harmonic Sums.
Eric Weisstein's World of Mathematics, Dirichlet eta function.

Cf. A069998, A094642.

RealDigits[Log[Sqrt[Pi/2]], 10, 105] // First
RealDigits[DirichletEta'[0], 10, 110][[1]] (* Eric W. Weisstein, Jan 06 2024 *)

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

Given the harmonic sum G(x) = Sum_{k>=1} (-1)^k*log(k)*exp(-k^2*x), lim_{x->0} G(x) = log(sqrt(Pi/2)).
Integral_{x=0..oo} G(x) dx = (Pi^2/12)*log(2) + zeta'(2)/2 = (Pi^2/12)*(EulerGamma + log(4*Pi) - 12*log(Glaisher)) = 0.1013165781635...
G'(0) = 7*zeta'(-2) = -7*zeta(3)/(4*Pi^2) = -0.2131391994...
Equals Integral_{-oo..+oo} -log(1/2 + i*z)/(exp(-Pi*z) + exp(Pi*z)) dz, where i is the imaginary unit. - Peter Luschny, Apr 08 2018
Equals Sum_{n>=0} Sum_{m>=1} (-1)^(m+n) * log(m+n)/(m+n) (Efthimiou, 2010). - Amiram Eldar, Apr 09 2022
Equals A094642/2. - R. J. Mathar, Jun 15 2023

A143149 Decimal expansion of 5sqrt(2Pi)/4.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Jul 27 2008

Upper bound using Shannon entropy arising in randomly-projected hypercubes.

			3.13328534328875...

David L. Donoho and Jared Tanner, Counting the faces of randomly-projected hypercubes and orthants, with applications, Discrete & computational geometry, Vol. 43 (2010), pp. 522-541, see p. 533; arXiv preprint, arXiv:0807.3590 [math.MG], 2008, see p. 10.
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, A217481, A019706, this sequence, A019710.

Cf. A143148 (lower bound).

RealDigits[5*Sqrt[2*Pi]/4, 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

5*sqrt(2*Pi)/4 \\ Michel Marcus, Mar 06 2020

Equals 10*Integral_{x>=0} x*sin(x^4) dx or 10*Integral_{x>=0} x*cos(x^4) dx (Fresnel integrals).

Edited and a(100) corrected by Georg Fischer, Jul 16 2021

A068141 Continued fraction for sqrt(Pi/2).

Original entry on oeis.org

1, 3, 1, 18, 9, 4, 1, 4, 3, 2, 1, 3, 1, 1, 2, 1, 5, 1, 2, 1, 1, 10, 66, 12, 111, 3, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 12, 6, 1, 47, 7, 1, 3, 1, 4, 1, 1, 45, 6, 15, 2, 11, 3, 102, 12, 1, 33, 1, 1, 1, 44, 3, 46, 1, 3, 35, 1, 6, 4, 3, 12, 1, 5, 2, 1, 4, 1, 3, 3

Offset: 0

Benoit Cloitre, Mar 13 2002

G. C. Greubel, Table of n, a(n) for n = 0..4999

Cf. A069998 (decimal expansion).

ContinuedFraction[Sqrt[Pi/2], 100] (* G. C. Greubel, Jan 12 2017 *)

contfrac(sqrt(Pi/2)) \\ Michel Marcus, Nov 23 2013

Offset changed by Andrew Howroyd, Aug 07 2024

A303617 Decimal expansion of Sum_{k >= 0} 2^(2k+1)/Product_{i = 0..k} (2i+1).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Apr 27 2018

			8.83943924091904909456698024436203574171002846378309279604186339401138107...
2/1 + 2^3/(1*3) + 2^5/(1*3*5) + 2^7/(1*3*5*7) + 2^9/(1*3*5*7*9) + 2^11/(1*3*5*7*9*11) + 2^13/(1*3*5*7*9*11*13) + ...

Cf. A001147, A004171.

Cf. A069998, A072334, A110894.

RealDigits[E^2 Sqrt[Pi/2] Erf[Sqrt[2]], 10, 100][[1]]

suminf(k=0, 2^(2*k+1)/prod(i=0, k, (2*i+1))) \\ Michel Marcus, Apr 27 2018

Equals e^2*sqrt(Pi/2)*erf(sqrt(2)) = A072334*A069998*A110894.

A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 22 2025

			0.49169967769382111771654625416890810022151027126875...

Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Solutions to the 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000, The Putnam Archive. See Problem A-4.
Editors of the Mathematics Magazine, The 61st Annual William Lowell Putnam Examination, News and Letters, Mathematics Magazine, Vol. 74, No. 1 (2001), pp. 75-83. See Problem A4, pages 77 and 79-80.
Jihyung Kang and others, Convergence of I = Integral_0^oo sinx sin(x2) dx, Mathematics StackExchange, 2017.
Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson, The Sixty-first William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 108, No. 9 (2001), pp. 841-850. See pages 843 and 846, Problem A4.
Eric Weisstein's World of Mathematics, Fresnel Integrals.
Wikipedia, Fresnel integral.

Cf. A069998, A217481, A231863.

RealDigits[Integrate[Sin[x]*Sin[x^2], {x, 0, Infinity}], 10, 120][[1]]
(* or *)
RealDigits[Sqrt[Pi/2] * (Cos[1/4] * FresnelC[1/Sqrt[2*Pi]] + Sin[1/4] * FresnelS[1/Sqrt[2*Pi]]), 10, 120][[1]]

Equals sqrt(Pi/2) * (cos(1/4) * FresnelC(1/sqrt(2*Pi)) + sin(1/4) * FresnelS(1/sqrt(2*Pi))), where FresnelC(x) and FresnelS(x) are the Fresnel integrals C(x) and S(x), respectively.

Equals Integral_{x=0..1/2} cos(x^2 - 1/4) dx.

cp's OEIS Frontend

A068465 Decimal expansion of Gamma(3/4).

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A218792 Decimal expansion of Sum_{n = -oo..oo} e^(-2*n^2).

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A244067 Decimal expansion of the Purdom-Williams constant, a constant related to the Golomb-Dickman constant and to the asymptotic evaluation of the expectation of a random function longest cycle length.

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

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A256358 Decimal expansion of log(sqrt(Pi/2)).

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A143149 Decimal expansion of 5*sqrt(2*Pi)/4.

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A068141 Continued fraction for sqrt(Pi/2).

A143149 Decimal expansion of 5sqrt(2Pi)/4.

A303617 Decimal expansion of Sum_{k >= 0} 2^(2k+1)/Product_{i = 0..k} (2i+1).