A002161 -id:A002161 - cp's OEIS frontend

A068470 Decimal expansion of exp(sqrt(Pi)).

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

Offset: 1

Benoit Cloitre, Mar 10 2002

			5.8852772500180288766117618534057698039906986189859...

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

Cf. A002161 (sqrt(Pi)), A039661 (exp(Pi)).

SetDefaultRealField(RealField(100)); R:=RealField(); Exp(Sqrt(Pi(R))); // G. C. Greubel, Nov 27 2018

evalf[120](exp(sqrt(Pi))); # Muniru A Asiru, Nov 28 2018

RealDigits[Exp[Sqrt[Pi]],10,120][[1]] (* Harvey P. Dale, Aug 22 2012 *)

default(realprecision, 100); exp(sqrt(Pi)) \\ G. C. Greubel, Jan 12 2017

numerical_approx(exp(sqrt(pi)), digits=100) # G. C. Greubel, Nov 27 2018

A088541 Decimal expansion of sqrt(Pi)/(2K)*exp(-gamma/2) where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Nov 16 2003

An illustration of the Chebyshev effect.

			0.868927768234323...

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 100.

Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See page 6.
S. Uchiyama, On some products involving primes, Proc. Amer. Math. Soc. 28 (1971) 629-630; MR 43#3227.

Cf. A002161, A064533, A088540, A155739.

digits = 104; LandauRamanujanK = 1/Sqrt[2]*NProduct[ ((1-2^(-2^n)) * Zeta[2^n] / DirichletBeta[2^n])^(1/2^(n+1)), {n, 1, 24}, WorkingPrecision -> digits+5]; Sqrt[Pi]/(2*LandauRamanujanK )*Exp[-EulerGamma/2] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 04 2013, updated Mar 14 2018 *)

Equals sqrt(Pi)/(2K)*exp(-gamma/2) = lim x-->oo prod(1-1/p) where p runs through the primes p==3 mod 4 and p<=x.

Equals A002161*A064533/(2*exp(A155739)). - Michel Marcus, Jun 19 2020

A093204 Decimal expansion of Pi^(-1/3).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.682784063255295681467020833...

Index entries for transcendental numbers

Cf. A000796, A002161, A049541, A197111.

RealDigits[Surd[Pi,-3],10,120][[1]] (* Harvey P. Dale, Aug 18 2014 *)

PARI
```
Pi^(-1/3) \\ M. F. Hasler, Oct 07 2014
```

1/A092039. - M. F. Hasler, Oct 07 2014

A249538 Decimal expansion of 3*sqrt(Pi), the average perimeter of a random Gaussian triangle in two dimensions.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 31 2014

			5.317361552716548081894502450023435548392648368367161...

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]
Eric Weisstein's MathWorld, Gaussian Triangle Picking
Index entries for transcendental numbers

Cf. A002161 (average side length in two dimensions), A102556, A102557.

RealDigits[3*Sqrt[Pi], 10, 105] // First

3*sqrt(Pi) \\ G. C. Greubel, Jan 09 2017

A294968 Decimal expansion of sqrt(7 + 4*sqrt(2))/2.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Nov 16 2017

A construction of this length using a circle of radius of 1 (length unit) and the circumscribed and inscribed square as well as a square in the middle between them is given in figure 15 of the footnote 13 on p. 26 of M. Gardner's book. Point A is on the middle of one side, say the left, of the middle square and point B is at the intersection of the prolonged right side of the inscribed square with the middle square. Then the length AB is sqrt(AC^2 + CB^2) with point C on the middle of the right side of the inscribed square. AC = (2 - sqrt(2))/4 and CB = 1/sqrt(2) + (2 - sqrt(2))/2 = (2 + sqrt(2))/4. Therefore, AB = sqrt(7 + 4*sqrt(2))/2. See the link.
This is a not a good approximation to the squaring of the circle problem: (AB)^2 is not Pi, or AB = 1.778... is not sqrt(Pi) = A002161 = 1.772...  Gardner writes that he was told "of an extremely good approximation".
An elegant construction for the reflexible Archimedean solids was devised by Alicia Boole Stott. In the process called expansion, certain sets of elements (i.e., edges or faces) are moved directly away from the center, retaining their size and orientation, until the consequent interstices can be filled with new regular faces. The reverse process is called contraction. The final edge length is the same as that of the starting solid. By contracting the truncated cube according to its triangles, the cuboctahedron is obtained. (Cf. W. W. Rouse Ball, H. S. M. Coxeter, Mathematical recreations and essays, pp. 139-140.) The factor of this contraction is 1/{a(n)} = (2/17)*sqrt(119-68*sqrt(2)) = 0.56216927542964050970... - Martin Renner, Dec 31 2019

			1.778823645663924450858334820415026760765017372952578...

Martin Gardner, Logic Machines and Diagrams, Second Ed., 1982, The Harvester Press, p. 26, Figure 15.
W. W. Rouse Ball, H. S. M. Coxeter, Mathematical recreations and essays, New York, Dover, 13th ed., 1987, pp. 139-140 (Mrs. Stott's Construction), fig. 3.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Alicia Boole Stott, Geometrical deduction of semiregular from regular polytopes and space fillings. In: Verhandelingen der Koninklijke Akademie van Wetenschappen, section 1, part 11, nr. 1. Amsterdam, Müller 1910, pp. 3-24.
Wolfdieter Lang, Some poor approximation of sqrt(Pi).

SetDefaultRealField(RealField(100)); Sqrt(7+4*Sqrt(2))/2; // G. C. Greubel, Sep 30 2018

RealDigits[Sqrt[7 + 4*Sqrt[2]]/2, 10, 100][[1]] (* G. C. Greubel, Sep 30 2018 *)

sqrt(7+4*sqrt(2))/2 \\ Felix Fröhlich, Nov 16 2017

A337092 Decimal expansion of sqrt(40/Pi).

Original entry on oeis.org

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

Offset: 1

Peter Munn, Aug 15 2020

A gauge point marked c^1 or c_1 ("c" with a superscripted or subscripted "1") on slide rule calculating devices in the 20th century. The Pickworth reference notes its use "in calculating the contents of cylinders".

			3.568248232305...

C. N. Pickworth, The Slide Rule, 24th Ed., Pitman, London, 1945, p. 53, Gauge Points.

Peter Munn, Aristo 89 Slide Rule
Index entries for transcendental numbers

Cf. A002161, A010494, A190732.

evalf(sqrt(40.0/Pi)) ; # R. J. Mathar, Sep 02 2020

RealDigits[Sqrt[40/Pi], 10, 100][[1]] (* Amiram Eldar, Aug 15 2020 *)

sqrt(40/Pi) \\ Michel Marcus, Aug 19 2020

Equals A010494/A002161 = 2*sqrt(10*A049541).

A131265 Decimal expansion of the negative of the first derivative of the Gamma Function at 1/2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Sep 28 2007

			3.4802309069132620269385951981443497500324293345037602151543...

G. C. Greubel, Table of n, a(n) for n = 1..10000
T. Amdeberhan, L. Medina and V. H. Moll, The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals, arXiv:0705.2379 [math.CA], 2007, example 3.1.

Cf. A020759, A002161.

RealDigits[ Sqrt[Pi]*PolyGamma[0, 1/2], 10, 59] // First (* Jean-François Alcover, Feb 20 2013 *)

PARI
```
print(sqrt(Pi)*(Euler+2*log(2)));
```

Equals A020759 * A002161.

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

A175575 Decimal expansion of (Gamma(3/4))^2 / Pi^(3/2) .

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 15 2010

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

			0.2696763005941896783339678...

Bruce C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. Lond. Math. Soc. vol 15 no 4 (1983) 273-320.

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

RealDigits[Gamma[3/4]^2/Pi^(3/2),10,120][[1]] (* Harvey P. Dale, Mar 16 2021 *)

Equals A068465^2 / (A000796 * A002161 ) = 1/A175576.

Equals (5/16)*hypergeom([1/4, -3/4], [3/2], 1). - Peter Bala, Mar 02 2022

A255405 a(n) = floor((2/sqrt(Pi))^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 23, 26, 29, 33, 37, 42, 47, 53, 60, 68, 77, 87, 98, 111, 125, 141, 159, 180, 203, 229, 258, 292, 329, 371, 419, 473, 534, 602, 680, 767, 865, 977, 1102, 1244, 1403, 1584, 1787, 2016, 2275, 2567

Offset: 0

Kival Ngaokrajang, Feb 22 2015

nonn

Inspired by squaring the circle and Vitruvian Man, but starting with a unit circle and a square whose sides are of length sqrt(Pi), A002161. a(n) is the curvature (rounded down) of the n-th circle. See illustrations in the links.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Kival Ngaokrajang, Illustration of initial terms.
Wikipedia, Squaring the circle.

Cf. A002161, A255162, A255163.

Table[Floor[(2/Sqrt[Pi])^n], {n,0,50}] (* G. C. Greubel, Jan 09 2017 *)

{for(n=1,100,a=floor(2^n/sqrt(Pi)^n);print1(a,", "))}

a(n) = floor((2/sqrt(Pi))^n).

cp's OEIS Frontend

A068470 Decimal expansion of exp(sqrt(Pi)).

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A088541 Decimal expansion of sqrt(Pi)/(2K)*exp(-gamma/2) where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

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A093204 Decimal expansion of Pi^(-1/3).

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A249538 Decimal expansion of 3*sqrt(Pi), the average perimeter of a random Gaussian triangle in two dimensions.

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A294968 Decimal expansion of sqrt(7 + 4*sqrt(2))/2.

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Magma

Mathematica

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A337092 Decimal expansion of sqrt(40/Pi).

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Maple

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PARI

Formula

A131265 Decimal expansion of the negative of the first derivative of the Gamma Function at 1/2.

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