A224268 -id:A224268 - cp's OEIS frontend

A179587 Decimal expansion of the volume of square cupola with edge length 1.

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.
Also, decimal expansion of 1 + Product_{n>0} (1-1/(4*n+2)^2). - Bruno Berselli, Apr 02 2013
Decimal expansion of 1 + (least possible ratio of the side length of one inscribed square to the side length of another inscribed square in the same non-obtuse triangle). - L. Edson Jeffery, Nov 12 2014
2*sqrt(2)/3 is the radius of the base of the maximum-volume right cone inscribed in a unit-radius sphere. - Amiram Eldar, Sep 25 2022

			1.942809041582063365867792482806465385713114583584632048784453158660...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Victor Oxman and Moshe Stupel, Why are the side lengths of the squares inscribed in a triangle so close to each other?, Forum Geometricorum, Vol. 13 (2013), 113-115.
Wolfram Alpha, Johnson solid 4
Index entries for algebraic numbers, degree 2

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A224268.

Cf. A131594 (decimal expansion of sqrt(2)/3).

RealDigits[N[1+(2*Sqrt[2])/3,200]]
(* From the second comment: *) RealDigits[N[1 + Product[1 - 1/(4 n + 2)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

sqrt(8)/3+1 \\ Charles R Greathouse IV, Nov 14 2016

Equals (3 + 2*sqrt(2))/3.

Equals 1 + 2*A131594. - L. Edson Jeffery, Nov 12 2014

A093341 Decimal expansion of "lemniscate case".

Original entry on oeis.org

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

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Apr 26 2004

			1.854074677301371918433850347195260046217598823521766905585928045056021...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, 1972, Section 18.14.7, p. 658.
Jonathan Borwein & Peter Borwein, A Dictionary of Real Numbers. Pacific Grove, California: Wadsworth & Brooks/Cole Advanced Books & Software, 1990, p. iii.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.1 Gauss' Lemniscate Constant, pp. 421-422.

Harry J. Smith, Table of n, a(n) for n = 1..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Section 18.14.7, p. 658.
G. Mingari Scarpello and D. Ritelli, On computing some special values of hypergeometric functions, arXiv:1212.0251 [math.CA], 2012-2014, eq. (4.1).
Eric Weisstein's World of Mathematics, Lemniscate Case.
I. J. Zucker and G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Camb. Phil. Soc. 131 (2001) 309-319, (2.3).

Cf. A064853, A062539, A105372, A153396, A175576, A224268.

evalf( EllipticK(1/sqrt(2)) ); # R. J. Mathar, Aug 28 2013

RealDigits[ N[ Gamma[1/4]^2 / (4*Sqrt[Pi]), 105]][[1]] (* Jean-François Alcover, Oct 04 2011 *)
RealDigits[N[EllipticK[1/2], 105]][[1]] (* Vaclav Kotesovec, Feb 22 2015 *)

{ allocatemem(932245000); default(realprecision, 5080); x=gamma(1/4)^2/(4*(Pi)^(1/2)); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b093341.txt", n, " ", d)); } \\ Harry J. Smith, Jun 19 2009

Pi/agm(sqrt(2),2) \\ Charles R Greathouse IV, Feb 04 2015

ellK(1/sqrt(2)) \\ Charles R Greathouse IV, Feb 04 2025

GAMMA(1/4)^2/(4*(Pi)^(1/2)). - Pab Ter (pabrlos(AT)yahoo.com), May 24 2004
Also equals ellipticK(1/sqrt(2)) = Pi/2*hypergeom([1/2,1/2],[1],1/2),
or also the smallest positive root of cs(x/sqrt(2)|-1), where cs is the Jacobi elliptic function, or also the real half-period of the Weierstrass Pe function (Cf. Finch p. 422). - Jean-François Alcover, Apr 30 2013, updated Aug 01 2014
From Peter Bala, Feb 22 2015: (Start)
Equals Integral_{x = 0..oo} 1/sqrt(1 + x^4) dx = 2 * Integral_{x = 0..1} 1/sqrt(1 + x^4) dx = sqrt(2) * Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.
Equals 2 * Sum {n >= 0} (-1/4)^n * binomial(2*n,n) * 1/(4*n + 1). (End)
Equals A062539 / sqrt(2). - Amiram Eldar, May 04 2022
Equals 1/A105372 = A175576/2 = 2*A224268. - Hugo Pfoertner, Aug 27 2024

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 24 2004

A096427 Decimal expansion of 1/(sqrt(2)*G), where G is Gauss's constant A014549.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2004

Also, decimal expansion of Product_{n>=1} (1-1/(4n-1)^2). - Bruno Berselli, Apr 02 2013

			0.8472130847939790866064991234821916364814459103269... = agm(1, sqrt(1/2))

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 1, equation 1:7:6 at page 13.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Peter Bala, Notes on the constants A096427 and A224268 .
Eric Weisstein's World of Mathematics, Gauss's Constant.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Ubiquitous Constant.
Index entries for transcendental numbers.

Cf. A014549, A062539, A224268, A091670 (1/C^2), A175574 (1/C), A293238 (C^2), A053004 (sqrt(2)*C), A327995.

SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(3/4)^2/(Sqrt(2)*Sqrt(Pi(R)/2)); // G. C. Greubel, Aug 17 2018

RealDigits[ArithmeticGeometricMean[1, Sqrt[2]]/Sqrt[2], 10, 110][[1]] (* Bruno Berselli, Apr 02 2013 *)
(* From the comment: *) RealDigits[N[Product[1 - 1/(4 n - 1)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

{ default(realprecision, 20080); x=agm(1, sqrt(1/2)); d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b096427.txt", n, " ", d)); } \\ Harry J. Smith, Apr 15 2009

agm(1, sqrt(1/2)) \\ Michel Marcus, Jun 09 2019

Also equals agm(1,1/sqrt(2)) since agm(1,1/b) = (1/b)*agm(1,b). - Gerald McGarvey, Sep 22 2008
From Peter Bala, Feb 26 2019: (Start)
C = Gamma(3/4)^2/sqrt(Pi).
C = 1/( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2.
C = (1/sqrt(2)) * 1/( Sum_{n = -inf..inf} (-1)^n*exp(-Pi*n^2 ) )^2.
Conjecturally, C = (1/sqrt(2)) * 1/( Sum_{n = -inf..inf} exp(-Pi*(n+1/2)^2 ) )^2.
C = ((-1)^m*4^m/binomial(2*m,m)) * Product_{n >= 0} ( 1 - (4*m + 1)^2/(4*n + 3)^2 ), for m = 0,1,2,....
C = 1 - Integral_{x = 0..1} (sqrt(1 + x^4) - 1)/x^2 dx.
C = 1 - Sum_{n >= 1} binomial(1/2,n)/(4*n - 1) = 1 - Sum_{n >= 0} (-1)^n/(4*n + 3)*Catalan(n)/2^(2*n + 1).
Continued fraction: 1 - 1/(3 + 6/(1 + 12/(3 + ... + (4*n - 1)*(4*n - 2)/(1 + 4*n*(4*n - 1)/(3 + ... ))))). (End)
From Peter Bala, Mar 02 2022 : (Start)
C = (2/3)*hypergeom([1/4, 3/4], [7/4], 1)
C = hypergeom([-1/4, 1/4], [3/4], 1).
C = hypergeom([-1/2, -1/4], [3/4], -1). Cf. A053004.
C = (16/21)*hypergeom([-1/4, -3/4], [7/4], 1). (End)
Equals Pi/(sqrt(2)*A062539). - Amiram Eldar, May 04 2022
C = Integral_{x = 0..Pi/2} sqrt(sin(x)*cos(x)) dx. - Adam Hugill, Nov 27 2022
Equals 1/A175574 = sqrt(A293238) = A327995^2. - Hugo Pfoertner, Dec 26 2024

A112628 Decimal expansion of 2*sqrt(2)/Pi.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jan 11 2006

Example of extension to Buffon's Needle Problem: The probability that the boundary of a square will intersect one of the parallel lines if the square's diagonal length l (almost) equals the distance d between each pair of lines. This follows directly from the Weisstein/MathWorld Buffon's Needle Problem link's statement P=p/(Pi*d), where P is the probability of intersection with any convex polygon's boundary if the generalized diameter of that polygon is less than d and p is the perimeter of the polygon. (Take d=l, then p=2*sqrt(2)*d.).

The area of a regular octagon circumscribed in a unit-area circle. - Amiram Eldar, Nov 05 2020

			0.9003163161571060695551991910067405826645741499552206255714374712314587307...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Buffon's needle problem.
Eric Weisstein's World of Mathematics, Generalized Diameter.
Index entries for transcendental numbers

Cf. A060294 (2/Pi), A089491 (3/Pi), A224268.

R:= RealField(100); 2*Sqrt(2)/Pi(R); // G. C. Greubel, Aug 17 2018

RealDigits[2 Sqrt[2]/Pi, 10, 110][[1]] (* Bruno Berselli, Apr 02 2013 *)
(* From the second comment: *) RealDigits[N[Product[1 - 1/(4 n)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

PARI
```
2*sqrt(2)/Pi
```

Equals Product_{n>=1} (1-1/(4*n)^2). - Bruno Berselli, Apr 02 2013
Equals sinc(Pi/4). - Peter Luschny, Oct 04 2019
Equals Product_{k>=3} cos(Pi/2^k). - Amiram Eldar, Aug 24 2020

A254794 Decimal expansion of L^2/Pi where L is the lemniscate constant A062539.

Original entry on oeis.org

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

Offset: 1

Peter Bala, Feb 22 2015

Brouncker gave the generalized continued fraction expansion 4/Pi = 1 + 1^2/(2 + 3^2/(2 + 5^2/(2 + ... ))). More generally, Osler shows that the continued fraction n + 1^2/(2*n + 3^2/(2*n + 5^2/(2*n + ... ))) equals a rational multiple of 4/Pi or its reciprocal when n is a positive odd integer, and equals a rational multiple of L^2/Pi or its reciprocal when n is a positive even integer.

			2.18843961522647663883676994070446454325937272282556672211928621....

O. Perron, Die Lehre von den Kettenbrüchen, Band II, Teubner, Stuttgart, 1957

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Peter Bala, Notes on the constants A096427 and A224268
B. C. Berndt, R. L. Lamphere and B. M. Wilson, Chapter 12 of Ramanujan's second notebook: Continued fractions, Rocky Mountain Journal of Mathematics, Volume 15, Number 2 (1985), 235-310.
T. J. Osler, The missing fractions in Brouncker's sequence of continued fractions for Pi, The Mathematical Gazette, 96(2012), pp. 221-225.

Cf. A000796, A062539, A254795, A254796, A096427, A133748, A224268.

SetDefaultRealField(RealField(110)); 2*(Gamma(5/4)/Gamma(3/2))^4; // G. C. Greubel, Mar 06 2019

#A254794
digits:=105:
2*( GAMMA(5/4)/GAMMA(3/2) )^4:
evalf(%);

RealDigits[2*(Gamma[5/4]/Gamma[3/2])^4, 10, 110][[1]] (* G. C. Greubel, Mar 06 2019 *)

default(realprecision, 110); 2*(gamma(5/4)/gamma(3/2))^4 \\ G. C. Greubel, Mar 06 2019

numerical_approx(2*(gamma(5/4)/gamma(3/2))^4, digits=110) # G. C. Greubel, Mar 06 2019

L^2/Pi = 2*( (1/4)!/(1/2)! )^4 = 9/4*( (1/4)!/(3/4)! )^2.
L^2/Pi = lim_{n -> oo} (4*n + 2) * Product {k = 0..n} ( (4*k - 1)/(4*k + 1) )^2
Generalized continued fraction: L^2/Pi = 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))). This is the particular case n = 0, x = 2 of a result of Ramanujan - see Berndt et al., Entry 25. See also Perron, p. 35.
The sequence of convergents to Ramanujan's continued fraction begins [2/1, 9/4, 54/25, 441/200, 4410/2025, ...]. See A254795 for the numerators and A254796 for the denominators.
Another continued fraction is L^2/Pi = 1 + 2/(1 + 1*3/(2 + 3*5/(2 + 5*7/(2 + 7*9/(2 + ... ))))), which can be transformed into the slowly converging series: L^2/Pi = 1 + 4 * Sum {n >= 0} P(n)^2/(4*n + 5), where P(n) = Product {k = 1..n} (4*k - 1)/(4*k + 1).
(L^2/Pi)^2 = 3 + 2*( 1^2/(1 + 1^2/(3 + 3^2/(1 + 3^2/(3 + 5^2/(1 + 5^2/(3 + ... )))))) ) follows by setting n = 0, x = 2 in Entry 26 of Berndt et al.
From Peter Bala, Feb 28 2019: (Start)
C = 2*A224268/A096427.
For m = 0,1,2,..., C = 4*(m + 1)*P(m)/Q(m), where P(m) = Product_{n >= 1} ( 1 - (4*m + 3)^2/(4*n + 1)^2 ) and Q(m) = Product_{n >= 0} ( 1 - (4*m + 1)^2/(4*n + 3)^2 ).
For m = 0,1,2,..., C = - Product_{k = 1..m} (1 - 4*k)/(1 + 4*k) * Product_{n >= 0} ( 1 - (4*m + 2)^2/(4*n + 1)^2 ) and
1/C = Product_{k = 0..m} (1 + 4*k)/(1 - 4*k) * Product_{n >= 0} ( 1 - (4*m + 2)^2/(4*n + 3)^2 ).
C = (Pi/2) * ( Sum_{n = -oo..oo} exp(-Pi*n^2) )^4. (End)
Equals A133748/Pi. - Hugo Pfoertner, Apr 13 2024

A371466 Decimal expansion of Product_{k>=1} (1 - 1/(3*k+1)^2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 31 2024

			0.8833193751427249786568447498242193512859342691...

Cf. A003881, A010503, A086089, A096427, A224268, A290570, A371467.

RealDigits[Gamma[1/3]^3/(4 Sqrt[3] Pi), 10, 100][[1]]

Equals Gamma(1/3)^3 / (4 * sqrt(3) * Pi).
Equals A290570/2. - Hugo Pfoertner, Mar 31 2024
Equals Integral_{x=0..1} (1-x^3)^(1/3) dx. - Mikhail Kurkov, Jun 29 2025

A371467 Decimal expansion of Product_{k>=0} (1 - 1/(3*k+2)^2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 31 2024

			0.6844634059797257270110769788663463289556838...

Cf. A003881, A010503, A086089, A096427, A224268, A224273, A371466.

RealDigits[(4/3) Pi^2/Gamma[1/3]^3, 10, 100][[1]]

Equals (4/3) * Pi^2 / Gamma(1/3)^3.

Equals 1/A224273. - Hugo Pfoertner, Mar 31 2024

cp's OEIS Frontend

A179587 Decimal expansion of the volume of square cupola with edge length 1.

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A093341 Decimal expansion of "lemniscate case".

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A096427 Decimal expansion of 1/(sqrt(2)*G), where G is Gauss's constant A014549.

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A112628 Decimal expansion of 2*sqrt(2)/Pi.

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A254794 Decimal expansion of L^2/Pi where L is the lemniscate constant A062539.

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A371466 Decimal expansion of Product_{k>=1} (1 - 1/(3*k+1)^2).

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