A014549 -id:A014549 - cp's OEIS frontend

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

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

A159570 Continued fraction for 1/(sqrt(2)*G), where G is Gauss's constant A014549.

Original entry on oeis.org

0, 1, 5, 1, 1, 5, 20, 1, 13, 2, 1, 1, 61, 1, 1, 2, 4, 1, 3, 1, 2, 1, 5, 1, 13, 1, 11, 7, 6, 2, 77, 7, 1, 5, 4, 8, 1, 1, 6, 4, 2, 1, 1, 2, 4, 1, 1, 2, 1, 3, 1, 1, 6, 6, 1, 7, 1, 10, 1, 1, 4, 1, 4, 2, 1, 7, 1, 4, 1, 2, 17, 2, 2, 1, 5, 2, 1, 2, 1, 1, 1, 3, 3, 1, 1, 6, 1, 1, 16, 3, 1320, 2, 2, 7, 5, 9, 1, 217, 3

Offset: 0

Harry J. Smith, Apr 16 2009

nonn

Continued fraction for A096427.

Also called the Ubiquitous Constant.

			0.847213084793979086606499123... = 0 + 1/(1 + 1/(5 + 1/(1 + 1/(1 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..20000
Eric Weisstein's World of Mathematics, Ubiquitous Constant.

Cf. A096427.

SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(2*Pi(R)^(3/2)/Gamma(1/4)^2); // G. C. Greubel, Sep 28 2018

ContinuedFraction[2*Pi^(3/2)/Gamma[1/4]^2, 100] (* G. C. Greubel, Sep 28 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(agm(1, sqrt(1/2))); for (n=0, 20000, write("b159570.txt", n, " ", x[n+1])); }

default(realprecision, 100); contfrac(2*Pi^(3/2)/gamma(1/4)^2) \\ G. C. Greubel, Sep 28 2018

A068466 Decimal expansion of Gamma(1/4).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Mar 10 2002

Nesterenko proves that this constant is transcendental (he cites Chudnovsky as the first to show this); in fact it is algebraically independent of Pi and e^Pi over Q. - Charles R Greathouse IV, Nov 11 2013

			3.6256099082219083119306851558676720029951676828800654674333...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:13 at page 414.

Harry J. Smith, Table of n, a(n) for n = 1..1000
William Duke and Özlem Imamoḡlu, Special values of multiple gamma functions, Journal de théorie des nombres de Bordeaux, Vol. 18. No. 1 (2006), pp. 113-123.
Greg J. Fee and Simon Plouffe, Gamma(1/4) to 25000 digits.
Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics, Vol. 187, No. 9 (1996), pp. 1319-1348. (English translation)
Simon Plouffe, Gamma(1/4) to 250000 digits.
Dan Romik, On Viazovska's modular form inequalities, arXiv:2303.13427 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Gamma Function.
Wikipedia, Particular values of the Gamma function: General rational arguments.
Index to sequences related to the Gamma function.
Index entries for transcendental numbers.

Cf. A014549, A248557.

R:= RealField(100); SetDefaultRealField(R); Gamma(1/4); // G. C. Greubel, Mar 10 2018

Maple
```
evalf(GAMMA(1/4));
```

RealDigits[Gamma[1/4], 10, 110][[1]] (* Bruno Berselli, Dec 13 2012 *)

default(realprecision, 1080); x=gamma(1/4); for (n=1, 1000, d=floor(x); x=(x-d)*10; write("b068466.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009

From Amiram Eldar, Jun 12 2021: (Start)
Equals sqrt(2*sqrt(2*Pi^3)*G), where G is Gauss's constant (A014549).
Equals (2*Pi)^(3/4) * Product_{k>=1} tanh(k*Pi/2) (Duke and Imamoḡlu, 2006). (End)
Gamma(1/4) * A068465 = A063448. - R. J. Mathar, May 22 2024
Equals Product_{n>=1} exp((2*(6*n + 1)*(1 - beta(n)) - (eta(n) - 1))/(4*n)), where eta(n) and beta(n) are the Dirichlet eta and beta functions, respectively. - Antonio Graciá Llorente, Sep 05 2024

A062539 Decimal expansion of the Lemniscate constant or Gauss's constant.

Original entry on oeis.org

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

Offset: 1

Jason Earls, Jun 25 2001

			2.622057554292119810464839589891119413682754951431623162816821703...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.3 and 6.2, pp. 99, 420.

Harry J. Smith, Table of n, a(n) for n = 1..5000
Lorenz Milla, World record computation of Lemniscate Constant (2,000,000,000,000 digits).
Simon Plouffe, Lemniscate or Gauss constant.
Simon Plouffe, Lemniscate constant or Gauss constant.
Eric Weisstein's World of Mathematics, Lemniscate Constant.
Wikipedia, Lemniscate constant.
Index entries for transcendental numbers.

Cf. A062540, A064853, A085565.
Cf. A002193, A014549, A093341.
Equals A000796/A053004 (see PARI script).

SetDefaultRealField(RealField(100)); R:= RealField(); (1/2)*Sqrt(2*Pi(R)^3)/Gamma(3/4)^2; // G. C. Greubel, Oct 07 2018

evalf((1/2)*sqrt(2*Pi^3)/GAMMA(3/4)^2,120); # Muniru A Asiru, Oct 08 2018
evalf(1/2*GAMMA(1/4)*GAMMA(1/2)/GAMMA(3/4),120); # Martin Renner, Aug 16 2019
evalf(1/2*Beta(1/4,1/2),120); # Martin Renner, Aug 16 2019
evalf(2*int(1/sqrt(1-x^4),x=0..1),120); # Martin Renner, Aug 16 2019

RealDigits[Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2, 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)

print(1/2*Pi^(3/2)/gamma(3/4)^2*2^(1/2))

allocatemem(932245000); default(realprecision, 5080); x=Pi^(3/2)*sqrt(2)/(2*gamma(3/4)^2); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062539.txt", n, " ", d)); \\ Harry J. Smith, Jun 20 2009

Pi/agm(1,sqrt(2)) \\ Charles R Greathouse IV, Feb 04 2015

intnum(x=0,Pi, 1/sqrt(1 + sin(x)^2)) \\ Charles R Greathouse IV, Feb 04 2025

Equals (1/2)*sqrt(2*Pi^3)/Gamma(3/4)^2.
A093341 multiplied by A002193. - R. J. Mathar, Aug 28 2013
From Martin Renner, Aug 16 2019: (Start)
Equals 2*Integral_{x=0..1} 1/sqrt(1-x^4) dx.
Equals 1/2*B(1/4,1/2) with Beta function B(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y). (End)
Equals Pi/AGM(1, sqrt(2)). - Jean-François Alcover, Feb 28 2021
Equals 2*hypergeom([1/2, 1/4], [5/4], 1). - Peter Bala, Mar 02 2022
Equals (1/2)*A064853 = 2*A085565. - Amiram Eldar, May 04 2022
Equals Pi*A014549. - Hugo Pfoertner, Jun 28 2024
Equals Integral_{x=0..Pi} 1/sqrt(1 + sin(x)^2) dx = EllipticK(-1) (see Finch at p. 420). - Stefano Spezia, Dec 15 2024
Equals Gamma(1/4)^2 / (sqrt(Pi)*2^(3/2)). - Vaclav Kotesovec, Apr 26 2025
Equals (161*6440^(1/4))/(2*Sum_{k>=0} N(k)/D(k)) with N(k) = Pochhammer(1/8,k) * Pochhammer(5/8,k) * (275+8640*k) and D(k) = (k!)^2*25921^k [Jorge Zuniga, 2023].

A053004 Decimal expansion of AGM(1,sqrt(2)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 21 2000

AGM(a,b) is the limit of the arithmetic-geometric mean iteration applied repeatedly starting with a and b: a_0=a, b_0=b, a_{n+1}=(a_n+b_n)/2, b_{n+1}=sqrt(a_n*b_n).

			1.19814023473559220743992249228...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 195.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, page 5.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 420.
J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Gauss's Constant.
Index entries for transcendental numbers.

Cf. A014549, A053002, A053003, A076390, A085565, A096427.

evalf(GaussAGM(1, sqrt(2)), 144);  # Alois P. Heinz, Jul 05 2023

RealDigits[ N[ ArithmeticGeometricMean[1, Sqrt[2]], 105]][[1]] (* Jean-François Alcover, Jan 30 2012 *)
RealDigits[N[(1+Sqrt[2])Pi/(4EllipticK[17-12Sqrt[2]]), 105]][[1]] (* Jean-François Alcover, Jun 02 2019 *)

default(realprecision, 20080); x=agm(1, sqrt(2)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b053004.txt", n, " ", d)) \\ Harry J. Smith, Apr 20 2009

2*real(agm(1, I)/(1+I)) \\ Michel Marcus, Jul 26 2018

from mpmath import mp, agm, sqrt
mp.dps=106
print([int(z) for z in list(str(agm(1, sqrt(2))).replace('.', '')[:-1])]) # Indranil Ghosh, Jul 11 2017

Equals Pi/(2*A085565). - Nathaniel Johnston, May 26 2011
Equals Integral_{x=0..Pi/2} sqrt(sin(x)) or Integral_{x=0..1} sqrt(x/(1-x^2)). - Jean-François Alcover, Apr 29 2013 [cf. Boros & Moll p. 195]
Equals Product_{n>=1} (1+1/A033566(n)) and also 2*AGM(1, i)/(1+i) where i is the imaginary unit. - Dimitris Valianatos, Oct 03 2016
Conjecturally equals 1/( Sum_{n = -infinity..infinity} exp(-Pi*(n+1/2)^2 ) )^2. Cf. A096427. - Peter Bala, Jun 10 2019
From Amiram Eldar, Aug 26 2020: (Start)
Equals 2 * A076390.
Equals Integral_{x=0..Pi} sin(x)^2/sqrt(1 + sin(x)^2) dx. (End)
Equals sqrt(2/Pi)*Gamma(3/4)^2 = Integral_{x = 0..1} 1/(1 - x^2)^(1/4) dx = Pi/Integral_{x = 0..1} 1/(1 - x^2)^(3/4) dx. - Peter Bala, Jan 05 2022
From Peter Bala, Mar 02 2022: (Start)
Equals 2*Integral_{x = 0..1} x^2/sqrt(1 - x^4) dx.
Equals 1 - Integral_{x = 0..1} (sqrt(1 - x^4) - 1)/x^2 dx.
Equals hypergeom([-1/2, -1/4], [3/4], 1) = 1 + Sum_{n >= 0} 1/(4*n + 3)*Catalan(n)*(1/2^(2*n+1)). Cf. A096427. (End)

More terms from James Sellers, Feb 22 2000

A244644 Consider the method used by Archimedes to determine the value of Pi (A000796). This sequence denotes the number of iterations of his algorithm which would result in a difference of less than 1/10^n from that of Pi.

Original entry on oeis.org

0, 1, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 25, 26, 28, 29, 31, 33, 34, 36, 38, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 68, 69, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 88, 89, 91, 93, 94, 96, 98, 99, 101, 103, 104, 106, 108, 109, 111, 113, 114

Offset: 0

William H. Richardson and Robert G. Wilson v, Jul 03 2014

It takes on average 5/3 iterations to yield another digit in the decimal expansion of Pi.
The side of a 96-gon inscribed in a unit circle is equal to sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3))))). This is the size of one of the two polygons that Archimedes used to derive that 3 + 10/70 < Pi < 3 + 10/71.
In the Mathematica program, we started with an inscribed triangle and a circumscribed triangle of a unit circle and used decimal precision to just over a 1000 places.
The perimeter of the circumscribed 3*2^n-polygon exceeds Pi by more than the deficit of the perimeter of the inscribed 3*2^n-polygon. If we were to give twice the weight of the inscribed 3*2^n-polygon to that of the circumscribed 3*2^n-polygon, then the convergence would be twice as fast!
From A.H.M. Smeets, Jul 12 2018: (Start)
Archimedes's scheme: set upp(0) = 2*sqrt(3), low(0) = 3 (hexagons); upp(n+1) = 2*upp(n)*low(n)/(upp(n)+low(n)) (harmonic mean, i.e., 1/upp(n+1) = (1/upp(n) + 1/low(n))/2), low(n+1) = sqrt(upp(n+1)*low(n)) (geometric mean, i.e., log(low(n+1)) = (log(upp(n+1)) + log(low(n)))/2), for n >= 0. Invariant: low(n) < Pi < upp(n); variant function: upp(n)-low(n) tends to zero for n -> inf. The error of low(n) and upp(n) decreases by a factor of approximately 4 each iteration, i.e., approximately 2 bits are gained by each iteration.
From Archimedes's scheme, set r(n) = (2*low(n) + upp(n))/3, r(n) > Pi and the error decreases by a factor of approximately 16 for each iteration, i.e., approximately 4 bits are gained by each iteration. This is often called "Snellius acceleration".
For similar schemes see also A014549 (in this case with quadratically convergence), A093954, A129187, A129200, A188615, A195621, A202541.
Note that replacing "5/3" by "log(10)/log(4)" would be better in the first comment. (End)

			Just averaging the initial two triangles (3.89711) would yield Pi to one place of accuracy, i.e., the single digit '3'. Therefore a(0) = 0.
The first iteration yields, as the perimeters of the two hexagons, 4*sqrt(3) and 6. Their average is ~ 3.2320508 which is within 1/10 of the true value of Pi. Therefore a(1) = 1.
a(3) = 5 since it takes 5 iterations of Archimedes's algorithm to drive the averaged value of the circumscribed 96-gon and the inscribed 96-gon to yield a value within 0.001 of the correct value of Pi.
a(4) = 6 since it takes 6 iterations of Archimedes's algorithm to drive the averaged value of the circumscribed 3*2^6-gon and the inscribed 3*2^6-gon to yield a value within 0.0001 of the correct value of Pi.

Petr Beckmann, A History of Pi, 5th Ed. Boulder, Colorado: The Golem Press (1982).
Jonathan Borwein and David Bailey, Mathematics by Experiment, Second Edition, A. K. Peters Ltd., Wellesley, Massachusetts 2008.
Jonathan Borwein & Keith Devlin, The Computer As Crucible, An Introduction To Experimental Mathematics, A. K. Peters, Ltd., Wellesley, MA, Chapter 7, 'Calculating [Pi]' pp. 71-79, 2009.
Eli Maor, The Pythagorean Theorem, Princeton Science Library, Table 4.1, page 55.
Daniel Zwillinger, Editor-in-Chief, CRC Standard Mathematical Tables and Formulae, 31st Edition, Chapman & Hall/CRC, Boca Raton, London, New York & Washington, D.C., 2003, §4.5 Polygons, page 324.

Mike Bertrand, Ex Libris, Archimedes and Pi
Frits Beukers and Weia Reinboud, Snellius versneld, (text in English), preprint.
Frits Beukers and Weia Reinboud, Snellius versneld, (text in English), NAW 5/3 no. 1, pp. 60-63 (2002).
Lee Fook Loong Eugene, The Computation of [Pi] And Its History
Kyutae Paul Han, Pi and Archimedes Polygon Method
Eric Weisstein's World of Mathematics, Archimedes' Recurrence Formula
Eric Weisstein's World of Mathematics, Regular Polygon
Michael Woltermann Ph.D., Washington & Jefferson College, 38. Archimedes' Determination of [Pi].

Cf. A000796.

a[n_] := a[n] = N[2 a[n - 1] b[n - 1]/(a[n - 1] + b[n - 1]), 2^10]; b[n_] := b[n] = N[ Sqrt[ b[n - 1] a[n]], 2^10]; a[-1] = 2Sqrt[27]; b[-1] = a[-1]/2; f[n_] := Block[{k = 0}, While[ 10^n*((a[k] + b[k])/4 -Pi) > 1, k++]; k]; Array[f, 70]

Conjecture: There exists a c such that a(n) = floor(n*log(10)/log(4)+c); where c is in the range [0.08554,0.10264]. Critical values to narrow the range are believed to be at a(74), a(133), a(192), a(251), a(310), a(366), a(425), a(484). - A.H.M. Smeets, Jul 23 2018

A053002 Continued fraction for 1 / M(1,sqrt(2)) (Gauss's constant).

Original entry on oeis.org

0, 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, 5, 2, 1, 1, 2, 2, 6, 9, 1, 1, 1, 3, 1, 2, 6, 1, 5, 1, 1, 2, 1, 13, 2, 2, 5, 1, 2, 2, 1, 5, 1, 3, 1, 3, 1, 2, 2, 2, 2, 8, 3, 1, 2, 2, 1, 10, 2, 2, 2, 3, 3, 1, 7, 1, 8, 3, 1, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 2, 17, 1, 4, 31, 2, 2, 5, 30, 1, 8, 2

Offset: 0

N. J. A. Sloane, Feb 21 2000

On May 30, 1799, Gauss discovered that this number is also equal to (2/Pi)*Integral_{t=0..1}(1/sqrt(1-t^4)).

M(a,b) is the limit of the arithmetic-geometric mean iteration applied repeatedly starting with a and b: a_0=a, b_0=b, a_{n+1}=(a_n+b_n)/2, b_{n+1}=sqrt(a_n*b_n).

			0.83462684167407318628142973...

J. M. Borwein and P. B. Borwein, Pi and the AGM, page 5.
J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

Harry J. Smith, Table of n, a(n) for n = 0..19999
Eric Weisstein's World of Mathematics, Gauss's Constant
G. Xiao, Contfrac
Index entries for continued fractions for constants
OEIS Wiki, Gauss's constant

Cf. A014549 (decimal expansion).

ContinuedFraction[1/ArithmeticGeometricMean[1, Sqrt[2]] , 100]  (* Jean-François Alcover, Apr 18 2011 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(1/agm(1, sqrt(2))); for (n=1, 20000, write("b053002.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009

More terms from James Sellers, Feb 22 2000

Offset changed by Andrew Howroyd, Aug 03 2024

A053003 Continued fraction for M(1,sqrt(2)).

Original entry on oeis.org

1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, 5, 2, 1, 1, 2, 2, 6, 9, 1, 1, 1, 3, 1, 2, 6, 1, 5, 1, 1, 2, 1, 13, 2, 2, 5, 1, 2, 2, 1, 5, 1, 3, 1, 3, 1, 2, 2, 2, 2, 8, 3, 1, 2, 2, 1, 10, 2, 2, 2, 3, 3, 1, 7, 1, 8, 3, 1, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 2, 17, 1, 4, 31, 2, 2, 5, 30, 1, 8, 2, 1

Offset: 0

N. J. A. Sloane, Feb 21 2000

M(a,b) is the limit of the arithmetic-geometric mean iteration applied repeatedly starting with a and b: a_0=a, b_0=b, a_{n+1}=(a_n+b_n)/2, b_{n+1}=sqrt(a_n*b_n).

			1.19814023473559220743992249228...

J. M. Borwein and P. B. Borwein, Pi and the AGM, page 5.
J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

Harry J. Smith, Table of n, a(n) for n = 0..19999
Eric Weisstein's World of Mathematics, Gauss's Constant.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A014549, A053002 without the leading term, A053004 (decimal expansion).

ContinuedFraction[ArithmeticGeometricMean[1,Sqrt[2]],100] (* Harvey P. Dale, Feb 26 2012 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(agm(1, sqrt(2))); for (n=1, 20000, write("b053003.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009

More terms from James Sellers, Feb 22 2000

Offset changed by Andrew Howroyd, Aug 03 2024

A358362 a(n) = 16^n * Sum_{k=0..n} (-1)^k*binomial(-1/2, k)^2.

Original entry on oeis.org

1, 12, 228, 3248, 56868, 846384, 14395920, 218556096, 3662534436, 56236646576, 933921124752, 14445103689408, 238434118702864, 3706773418885824, 60917716297733184, 950622015752780544, 15571249887287040804, 243694280206569964464, 3981466564018425521424

Offset: 0

Peter Luschny, Nov 12 2022

nonn

Cf. A358363, A358364, A358365, A367330.

a := n -> 16^n*add((-1)^k*binomial(-1/2, k)^2, k = 0..n):
seq(a(n), n = 0..19);

a[n_] := 16^n * Sum[(-1)^k*Binomial[-1/2, k]^2, {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 12 2022 *)

a(n) = (64*(2*n - 1)^2*a(n - 2) + (16*n - 4)*a(n - 1)) / n^2.
G.f.: hypergeom([1/2, 1/2], [1], -16*x)/(16*x - 1).
G.f.: 2*EllipticK(4*I*sqrt(x))/(Pi*(1 - 16*x)).
a(n) ~ A014549 * 2^(4*n). - Vaclav Kotesovec, Nov 14 2023

A222882 Decimal expansion of Sierpiński's second constant, K2 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} A004018(i^2)) - 4/Pi * log(n)).

Original entry on oeis.org

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

Offset: 1

Ant King, Mar 11 2013

Sierpiński introduced three constants in his 1908 doctoral thesis. The first, K, is very well known, bears his name and its decimal expansion is given in A062089. However, the second and third of these constants appear to have been largely forgotten. This sequence gives the decimal expansion of the second one, K2, and A222883 gives the decimal expansion of the third , K3. The formula given below show that K2 is related to several other, naturally occurring constants.

			K2 = 2.25492246288826476626818475952872355787166159860535188913831...

Steven R. Finch, Mathematical Constants, Encyclopaedia of Mathematics and its Applications, Cambridge University Press (2003), p.123. Corrigenda in the link below.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, (June 2012), pp. 15-16.
A. Schinzel, Wacław Sierpiński’s papers on the theory of numbers, Acta Arithmetica XXI, (1972), pp. 7-13. Corrigenda in "Dzieje Matematyki Polskiej" (Wrocław 2012), p.228 (in Polish).

Cf. A001620, A004018, A014549, A062089, A062539, A074962, A222883.

Take[Flatten[RealDigits[N[4(12 Log[Gamma[3/4]]-9 Log[Pi]+72 Log[Glaisher]-5 Log[2]+3 EulerGamma-3)/(3 Pi),100]]],86]

4/Pi*(log(exp(3*Euler-1)/(2^(2/3)/agm(sqrt(2),1)^2)) - 12/Pi^2*zeta'(2)) \\ Charles R Greathouse IV, Dec 12 2013

K2 = 4 / Pi * (eulergamma + K / Pi - 12 / Pi^2 * zeta'(2) + log(2) / 3 -1), where K  is Sierpiński's first constant (A062089) and eulergamma is the Euler-Mascheroni constant (A001620).
K2 = 4 * (12 * log(Gamma(3/4)) - 9*log(Pi) + 72*log(A) - 5*log(2) + 3 * eulergamma - 3) / (3 * Pi), where A is the Glaisher-Kinkelin constant (A074962).
K2 = 4 * (12 * log(Gamma(3/4)) + log(A^72 * e^(3*eulergamma - 3) / (32 * Pi^9))) / (3 * Pi).
K2 = 4 / Pi * (log(e^(3*eulergamma - 1) / (2^(2/3) * G^2)) - 12 / Pi^2 * zeta'(2)), where G is Gauss’ AGM constant (A014549).
K2 = 4 / Pi * (log(Pi^2 * e^(3*eulergamma - 1) / (2^(2/3) * L^2)) - 12 / Pi^2 * zeta'(2)), where L is Gauss’ lemniscate constant (A062539).

Minor edits by Vaclav Kotesovec, Nov 14 2014

cp's OEIS Frontend

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

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A159570 Continued fraction for 1/(sqrt(2)*G), where G is Gauss's constant A014549.

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A068466 Decimal expansion of Gamma(1/4).

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A062539 Decimal expansion of the Lemniscate constant or Gauss's constant.

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A053004 Decimal expansion of AGM(1,sqrt(2)).

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A244644 Consider the method used by Archimedes to determine the value of Pi (A000796). This sequence denotes the number of iterations of his algorithm which would result in a difference of less than 1/10^n from that of Pi.

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