A068521 -id:A068521 - cp's OEIS frontend

A084895 Decimal expansion of agm(1, 3), the arithmetic-geometric mean of 1 and 3.

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

Offset: 1

Eric W. Weisstein, Jun 10 2003

			1.8636167832448965423556890310242705951575328568268537222204412269783257945...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean.
Wikipedia, Arithmetic-geometric mean.

Cf. A068521 (agm(1,2)), A084896 (agm(1,4)), A084897 (agm(1,5)), A000796, A249282.

evalf(GaussAGM(1, 3), 144);  # Alois P. Heinz, Jul 04 2023

RealDigits[ArithmeticGeometricMean[1, 3], 10, 120][[1]] (* Vincenzo Librandi, Mar 12 2015 *)
RealDigits[N[Pi/EllipticK[1/4], 102]][[1]] (* Jean-François Alcover, Jun 02 2019 *)

agm(1, 3) \\ Michel Marcus, Jun 02 2019

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

Equals Pi/EllipticK(1/4) = A000796 / A249282. - Amiram Eldar, Apr 28 2025

A084896 Decimal expansion of agm(1,4), the arithmetic-geometric mean of 1 and 4.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jun 10 2003

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean

Cf. A068521 (agm(1,2)), A084895 (agm(1,3)), A084897 (agm(1,5)).

RealDigits[ArithmeticGeometricMean[1,4],10,120][[1]] (* Harvey P. Dale, May 25 2014 *)
RealDigits[N[5Pi/(4EllipticK[9/25]), 102]][[1]] (* Jean-François Alcover, Jun 02 2019 *)

PARI
```
agm(1, 4) \\ Michel Marcus, Jun 02 2019
```

2.2430285802876025701278021928290665405089731424066099759188237013874048...

A084897 Decimal expansion of agm(1,5), the arithmetic-geometric mean of 1 and 5.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jun 10 2003

			2.6040081905309402886964274487250343309270652556376494877484132544556792805685...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean

Cf. A068521 (agm(1,2)), A084895 (agm(1,3)), A084896 (agm(1,4)).

RealDigits[ArithmeticGeometricMean[1, 5], 10, 120][[1]] (* Vincenzo Librandi, Mar 12 2015 *)
RealDigits[N[3Pi/(2EllipticK[4/9]), 102]][[1]] (* Jean-François Alcover, Jun 02 2019 *)

PARI
```
agm(1, 5) \\ Michel Marcus, Jun 02 2019
```

A370543 Expansion of the Jacobi elliptic function cn(x,k) at k = 2 (even powers only).

Original entry on oeis.org

1, -1, 17, -433, 20321, -1584289, 179967473, -28151779537, 5812048858049, -1529741412486721, 499975227342256337, -198676311845589783793, 94327947921149101192481, -52736138158762405338195169, 34291374178966525773142501553, -25660133983889999165774819970577

Offset: 0

Paul D. Hanna, Mar 25 2024

sign

			E.g.f.: C(x) = 1 - x^2/2! + 17*x^4/4! - 433*x^6/6! + 20321*x^8/8! - 1584289*x^10/10! + 179967473*x^12/12! - 28151779537*x^14/14! + ...
where C(x) = cn(x,2).

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

Paul D. Hanna, Table of n, a(n) for n = 0..301

Cf. A028296 (cn(x,1)), A060627 (cn(x,k)).

Cf. A370542 (sn(x,2)), A370544 (dn(x,2)), A249282.

# a(n) = (2*n)! * [x^(2*n)] cn(x, 2).
cn_list := proc(k, len) local n; seq((2*n)!*coeff(series(JacobiCN(z, k), z,
2*len + 2), z, 2*n), n = 0..len) end:
cn_list(2, 15);  # Peter Luschny, Mar 25 2024

nmax = 20;
DeleteCases[CoefficientList[JacobiCN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax])! (* Jean-François Alcover, Mar 28 2024 *)

/* C(x) = Jacobi Elliptic Function cn(x,k) at k = 2: */
{a(n) = my(k=2,C=1,S=x,D=1); for(i=1,n,
S = intformal(C*D + x*O(x^(2*n+1)));
C = 1 - intformal(S*D);
D = 1 - k^2*intformal(S*C)); (2*n)!*polcoeff(C,2*n)}
for(n=0,20,print1(a(n),", "))

a(n) = (-1)^n * Sum_{k=0..n-1} A060627(n,k)*4^k for n >= 1, with a(0) = 1.
E.g.f. C(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
(1) C(x) = cn(x,k) at k = 2.
(2.a) C(x) = dn(2*x, 1/2).
(2.b) C(x) = (4 - 2*sn(x,1/2)^2 + sn(x,1/2)^4) / (4 - sn(x,1/2)^4).
(3) C(x) = 1 - Integral sqrt(1 - C(x)^2) * sqrt(4*C(x)^2 - 3) dx.
(4) C(x) = cos( Integral sqrt(4*C(x)^2 - 3) dx ).
(5.a) C(x) = sqrt(1 - sn(x,2)^2).
(5.b) C(x) = sqrt(3 + dn(x,2)^2) / 2.
O.g.f.: 1/(1 + x/(1 + 4*2^2*x/(1 + 3^2*x/(1 + 4*4^2*x/(1 + 5^2*x/(1 + 4*6^2*x/(1 + 7^2*x/(1 + ...)))))))) = 1 - x + 17*x^2 - 433*x^3 + 20321*x^4 - 1584289*x^5 + ... (continued fraction, see Wall, 94.18, p. 374). - [See formula in A060627 by Peter Bala, Apr 25 2017].
a(n) ~ (-1)^n * 2^(4*n+2) * agm(1,2)^(2*n+1) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)), where agm(1,2) = A068521 is the arithmetic-geometric mean. - Vaclav Kotesovec, Mar 28 2024

A249283 Decimal expansion of K(3/4), where K is the complete elliptic integral of the first kind.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 24 2014

			2.15651564749964323543867499880032202886411021649282536...

Steven R. Finch, Gergonne-Schwarz Surface, April 12, 2013. [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.

Cf. A093341 (K(1/2)), A249282 (K(1/4)), A000796, A068521.

evalf(EllipticK(sqrt(3)/2), 120); # Vaclav Kotesovec, Apr 22 2015

RealDigits[EllipticK[3/4], 10, 102] // First

ellK(sqrt(3/4)) \\ Charles R Greathouse IV, Feb 04 2025

Equals Pi/agm(1, 2) = A000796 / A068521. - Amiram Eldar, Apr 28 2025

A370542 Expansion of the Jacobi elliptic function sn(x,k) at k = 2 (odd powers only).

Original entry on oeis.org

1, -5, 73, -2765, 171409, -16145045, 2168436697, -391723265885, 91633164775201, -26955095234906405, 9737498127795037033, -4237907290209405609965, 2187044171819241257792689, -1320533769141977996485790645, 922274662722967857470247551737, -737730926392606318468533810754685

Offset: 0

Paul D. Hanna, Mar 23 2024

sign

			E.g.f.: S(x) = x - 5*x^3/3! + 73*x^5/5! - 2765*x^7/7! + 171409*x^9/9! - 16145045*x^11/11! + 2168436697*x^13/13! - 391723265885*x^15/15! + ...
where S(x) = sn(x,2).

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

Paul D. Hanna, Table of n, a(n) for n = 0..301

Cf. A000182 (unsigned sn(x,1)), A060628 (sn(x,k)).

Cf. A370543 (cn(x,2)), A370544 (dn(x,2)), A249282.

# a(n) = (2*n+1)! * [x^(2*n+1)] sn(x, 2).
sn_list := proc(k, len) local n; seq((2*n+1)!*coeff(series(JacobiSN(z, k), z,
2*len + 2), z, 2*n + 1), n = 0..len) end:
sn_list(2, 15);  # Peter Luschny, Mar 25 2024

nmax = 20;
DeleteCases[CoefficientList[JacobiSN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax] + 1)! (* Jean-François Alcover, Mar 28 2024 *)

/* S(x) = Jacobi Elliptic Function sn(x,k) at k = 2: */
{a(n) = my(S, k = 2); S = serreverse( intformal( 1/sqrt((1-x^2)*(1-k^2*x^2 +x*O(x^(2*n+2)) ) ) ));
(2*n+1)!*polcoeff(S,2*n+1)}
for(n=0,20, print1( a(n), ", ") );

a(n) = (-1)^n * Sum_{k=0..n} A060628(n,k)*4^k for n >= 0.
E.g.f. S(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
(1) S(x) = sn(x,k) at k = 2.
(2.a) S(x) = sn(2*x,1/2)/2.
(2.b) S(x) = sn(x,1/2) * cn(x,1/2) * dn(x,1/2) / (1 - sn(x,1/2)^4/4).
(3.a) S(x) = Series_Reversion( Integral 1/sqrt( (1-x^2)*(1-4*x^2) ) dx ).
(3.b) S(x) = Integral sqrt(1 - S(x)^2) * sqrt(1 - 4*S(x)^2) dx.
(4.a) S(x) = sin( Integral sqrt(1 - 4*S(x)^2) dx ).
(4.b) S(x) = sin( 2 * Integral sqrt(1 - S(x)^2) dx ) / 2.
(5.a) S(x) = sqrt(1 - cn(x,2)^2).
(5.b) S(x) = sqrt(1 - dn(x,2)^2) / 2.
O.g.f.: x/(1 + 5*x - 4*1*2^2*3*x^2/(1 + 5*3^2*x - 4*3*4^2*5*x^2/(1 + 5*5^2*x - 4*5*6^2*7*x^2/(1 + 5*7^2*x - 4*7*8^2*9*x^2/(1 + 5*9^2*x - ...))))) = x - 5*x^2 + 73*x^3 - 2765*x^4 + 171409*x^5 - 16145045*x^6 + ... (continued fraction, see Wall, 94.17, p. 374).
a(n) ~ (-1)^n * 2^(4*n+4) * agm(1,2)^(2*n+2) * n^(2*n + 3/2) / (Pi^(2*n + 3/2) * exp(2*n)), where agm(1,2) = A068521 is the arithmetic-geometric mean. - Vaclav Kotesovec, Mar 28 2024

A370544 Expansion of the Jacobi elliptic function dn(x,k) at k = 2 (even powers only).

Original entry on oeis.org

1, -4, 32, -832, 41216, -3168256, 359518208, -56319950848, 11624409595904, -3059387770077184, 999955757611876352, -397353151288859164672, 188655750511199441125376, -105472284295853235792510976, 68582751548430569936978444288, -51320267059211655419226235076608

Offset: 0

Paul D. Hanna, Mar 25 2024

sign

			E.g.f.: D(x) = 1 - 4*x^2/2! + 32*x^4/4! - 832*x^6/6! + 41216*x^8/8! - 3168256*x^10/10! + 359518208*x^12/12! - 56319950848*x^14/14! + ...
where D(x) = dn(x,2).

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

Paul D. Hanna, Table of n, a(n) for n = 0..301

Cf. A028296 (dn(x,1)), A060627 (cn(x,k)).

Cf. A370542 (sn(x,2)), A370543 (cn(x,2)), A249282.

# a(n) = (2*n)! * [x^(2*n)] dn(x, 2).
dn_list := proc(k, len) local n; seq((2*n)!*coeff(series(JacobiDN(z, k), z,
2*len + 2), z, 2*n), n = 0..len) end:
dn_list(2, 15);  # Peter Luschny, Mar 25 2024

nmax = 20;
DeleteCases[CoefficientList[JacobiDN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax])! (* Jean-François Alcover, Mar 28 2024 *)

/* D(x) = Jacobi Elliptic Function dn(x,k) at k = 2: */
{a(n) = my(k=2, C=1,S=x,D=1); for(i=1,n,
S = intformal(C*D + x*O(x^(2*n+1)));
C = 1 - intformal(S*D);
D = 1 - k^2*intformal(S*C)); (2*n)!*polcoeff(D,2*n)}
for(n=0,20,print1(a(n),", "))

a(n) = (-1)^n * Sum_{k=0..n-1} A060627(n,k)*4^(n-k) for n >= 1, with a(0) = 1.
E.g.f. D(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
(1) D(x) = dn(x,k) at k = 2.
(2.a) D(x) = cn(2*x, 1/2).
(2.b) D(x) = (4 - 8*sn(x,1/2)^2 + sn(x,1/2)^4) / (4 - sn(x,1/2)^4).
(3) D(x) = 1 - Integral sqrt(1 - D(x)^2) * sqrt(3 + D(x)^2) dx.
(4) D(x) = cos( Integral sqrt(3 + D(x)^2) dx ).
(5.a) D(x) = sqrt(1 - 4*sn(x,2)^2).
(5.b) D(x) = sqrt(4*cn(x,2)^2 - 3).
O.g.f. 1/(1 + 4*x/(1 + 2^2*x/(1 + 4*3^2*x/(1 + 4^2*x/(1 + 4*5^2*x/(1 + 6^2*x/(1 + 4*7^2*x/(1 + ...)))))))) = 1 - 4*x + 32*x^2 - 832*x^3 + 41216*x^4 - 3168256*x^5 + ... (continued fraction, see Wall, 94.19, p. 374).
a(n) ~ (-1)^n * 2^(4*n+3) * agm(1,2)^(2*n+1) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)), where agm(1,2) = A068521 is the arithmetic-geometric mean. - Vaclav Kotesovec, Mar 28 2024

A309893 Decimal expansion of AGM(1, sqrt(3)/2).

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Aug 21 2019

Related to the pendulum acceleration relation at 60 degrees. In general, the period T of a mathematical pendulum with a maximum deflection angle theta is 2*Pi*sqrt(L/g)/AGM(1, cos(theta/2)), where L is the length of the pendulum, g is the gravitational acceleration, and 0 < theta <= 90 degrees. For theta = 60 degrees, the period is T = 2*Pi*sqrt(L/g)/AGM(1, sqrt(3)/2). - Jianing Song, Nov 21 2022

			0.931808391622448271177844...

Wikipedia, Pendulum (mechanics)

Cf. A310000 (AGM(1, cos(Pi/5))), A096427 (AGM(1, sqrt(2)/2)), A053004, A014549, A068521.

RealDigits[ArithmeticGeometricMean[1, Sqrt[3]/2], 10, 100][[1]] (* Amiram Eldar, Aug 21 2019 *)

agm(1, sqrt(3)/2) \\ Michel Marcus, Aug 22 2019

import decimal
prec = int(input('Precision: '))
decimal.getcontext().prec = prec
D = decimal.Decimal
def agm(a, b):
    for x in range(prec):
        a, b = (a + b) / 2,(a * b).sqrt()
    return a
print(agm(1, D(3).sqrt()/2))

RealField(300)(1.0).agm(sqrt(3)/2) # Peter Luschny, Aug 22 2019

AGM(1, sin(Pi/3)).

A310000 Decimal expansion of AGM(1, phi/2), where phi is the golden ratio (A001622).

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Aug 26 2019

Related to the pendulum acceleration relation at 72 degrees. 2*Pi*sqrt(l/g)/AGM(1, phi/2) gives the period T of a mathematical pendulum with a maximum deflection angle of 72 degrees from the downward vertical. The length of the pendulum is l and g is the gravitational acceleration.

			0.9019793381143431233972715365...

Cf. A001622, A309893, A053004, A014549, A068521.

RealDigits[ArithmeticGeometricMean[1, GoldenRatio/2], 10, 100][[1]] (* Amiram Eldar, Aug 26 2019 *)

agm(1, cos(Pi/5)) \\ Michel Marcus, Apr 05 2020

import decimal
iters = int(input('Precision: '))
decimal.getcontext().prec = iters
D = decimal.Decimal
def agm(a, b):
    for x in range(iters):
        a, b = (a + b) / 2,(a * b).sqrt()
    return a
print(agm(1, (D(5).sqrt()+1)/4))

Equals AGM(1, cos(Pi/5)).

A163766 Continued fraction expansion of the arithmetic-geometric mean of 1 and 2.

Original entry on oeis.org

1, 2, 5, 3, 2, 167, 1, 1, 99, 7, 10, 2, 2, 2, 2, 4, 1, 264, 2, 7, 2, 2, 2, 13, 1, 9, 3, 1, 4, 1, 1, 2, 15, 3, 1, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 4, 2, 1, 1, 5, 1, 1, 2, 16, 1, 5, 3, 1, 3, 2, 3, 1, 1, 2, 1, 43, 3, 1, 30, 1, 3, 153, 1, 1, 2, 1, 3, 1, 1, 13, 4, 6, 1, 1, 11, 2, 1, 2, 98, 1, 3, 1, 3, 1, 5, 15, 4, 1

Offset: 0

B. R. Becker (bbecker(AT)panda3.phys.unm.edu), Aug 03 2009

			This is the continued fraction of AGM(1,2) = 1.4567910310....

G. C. Greubel, Table of n, a(n) for n = 0..4999
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean
Eric Weisstein's World of Mathematics, Continued Fraction

Cf. A068521 (decimal expansion).

ContinuedFraction[ArithmeticGeometricMean[1, 2], 98]

Offset changed by Andrew Howroyd, Aug 09 2024

cp's OEIS Frontend

A084895 Decimal expansion of agm(1, 3), the arithmetic-geometric mean of 1 and 3.

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A084896 Decimal expansion of agm(1,4), the arithmetic-geometric mean of 1 and 4.

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A084897 Decimal expansion of agm(1,5), the arithmetic-geometric mean of 1 and 5.

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A370543 Expansion of the Jacobi elliptic function cn(x,k) at k = 2 (even powers only).

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A249283 Decimal expansion of K(3/4), where K is the complete elliptic integral of the first kind.

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A370542 Expansion of the Jacobi elliptic function sn(x,k) at k = 2 (odd powers only).

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A370544 Expansion of the Jacobi elliptic function dn(x,k) at k = 2 (even powers only).

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