A053004 -id:A053004 - cp's OEIS frontend

A071004 Binary expansion of AGM(1,sqrt(2)) where AGM(x,y) denote the arithmetic-geometric mean of (x,y).

1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Benoit Cloitre, Aug 17 2002

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A053004.

RealDigits[ArithmeticGeometricMean[1,Sqrt[2]],2,120][[1]] (* Harvey P. Dale, Mar 30 2015 *)

a(n)=agm(sqrt(2),1)*2^(n-1)\1%2 \\ Charles R Greathouse IV, Mar 03 2016

A248589 Decimal expansion of I, a constant appearing (as I^2) in the asymptotic variance of the area of the convex hull of random points in the unit square.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 09 2014

			1.061824136490969662805378287398947131005559644732889212...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.1 Geometric probability constants, p. 481.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's MathWorld, Square Point Picking

Cf. A053004, A085565, A096428, A096429.

RealDigits[Sqrt[2]*Pi^2/Gamma[1/4]^2, 10, 100][[1]]

sqrt(2)*Pi^2/gamma(1/4)^2 \\ G. C. Greubel, Jun 02 2017

I = sqrt(Pi/8)*(2-integral_{1..infinity} (sqrt(1+s^2)-s)*s^(-3/2) ds).
I = sqrt(Pi/2)*A053004, where A053004 is the arithmetic-geometric mean of 1 and sqrt(2).
I = Pi^(3/2)/(4*A085565), where A085565 is the lemniscate constant A.
I = sqrt(2)*Pi^2/Gamma(1/4)^2.

A269430 Decimal expansion of (1 + Pi)/2.

Original entry on oeis.org

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

Offset: 1

Jani Melik, Feb 26 2016

			2.0707963267948966192313216916397514420985846996875529...

Index entries for transcendental numbers

Cf. A000796, A002193, A002194, A002163, A174968, A001622, A002161, A010767, A011002, A011003, A191502, A053004.

PARI
```
(1 + Pi)/2 \\ Altug Alkan, Apr 07 2016
```
Sage
```
N((1+pi)/2, digits=110)
```

Equals A096444 + 1 or A019669 + 1/2.

Equals Sum_{k>=0} 2^k/binomial(2*k+2,k). - Amiram Eldar, Jun 30 2020

A275322 Decimal expansion of AGM(1, sqrt(2))^2/Pi.

Original entry on oeis.org

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

Offset: 0

Dimitris Valianatos, Jul 23 2016

Conjecture: Equals Product_{n odd} (n/(n+2) if n == 1 (mod 4), (n+2)/n otherwise) = (1/3) * (5/3) * (5/7) * (9/7) * (9/11) * (13/11) * (13/15) * (17/15) * (17/19) * (21/19) * (21/23) * (25/23) * (25/27) * ...

			0.45694658104446362537496662254768...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 21.

Cf. A053004 (AGM(1, sqrt(2))).

Cf. A000796, A002161, A068466, A212002.

SetDefaultRealField(RealField(100)); R:= RealField(); 8*Pi(R)^2/Gamma(1/4)^4; // G. C. Greubel, Oct 07 2018

evalf(GaussAGM(1,sqrt(2))^2/Pi,100); # Muniru A Asiru, Oct 08 2018

First@ RealDigits@ N[ArithmeticGeometricMean[1, Sqrt[2]]^2/Pi, 120] (* Michael De Vlieger, Jul 26 2016 *)

PARI
```
agm(1, sqrt(2)) ^ 2 / Pi
```

8*Pi^2/gamma(1/4)^4 \\ Altug Alkan, Oct 08 2018

Equals 8*Pi^2/Gamma(1/4)^4 = 4*Gamma(3/4)^2/Gamma(1/4)^2. - Vaclav Kotesovec, Sep 22 2016

A172084 Decimal expansion of the constant x that satisfies Arithmetic-Geometric-Mean(3,x) = Pi.

Original entry on oeis.org

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

Offset: 1

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 25 2010

			AGM(3,3.28645055277941042287...) = Pi = A000796.

Gerd Lamprecht, Iterationsrechner mit Algorithmus.
Gerd Lamprecht, Zahlenfolgen (sequences).
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean.

Cf. A000796, A053004.

RealDigits[x /. FindRoot[ArithmeticGeometricMean[3, x] == Pi, {x, 4}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, May 25 2023 *)

solve(x=3,4,agm(3,x)-Pi) \\ Charles R Greathouse IV, Mar 03 2016

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A071004 Binary expansion of AGM(1,sqrt(2)) where AGM(x,y) denote the arithmetic-geometric mean of (x,y).

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A248589 Decimal expansion of I, a constant appearing (as I^2) in the asymptotic variance of the area of the convex hull of random points in the unit square.

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A269430 Decimal expansion of (1 + Pi)/2.

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A275322 Decimal expansion of AGM(1, sqrt(2))^2/Pi.

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A172084 Decimal expansion of the constant x that satisfies Arithmetic-Geometric-Mean(3,x) = Pi.

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