A249282 -id:A249282 - 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

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

A338004 Decimal expansion of the angle of association yielding the gyroid relative to Schwarz's D surface.

Original entry on oeis.org

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

Offset: 0

Jeremy Tan, Oct 06 2020

For every minimal surface, an associate family of minimal surfaces can be defined by adding an angle of association to the base surface's Weierstrass-Enneper parametrization.

If the base is Schwarz's D surface, an angle of association of Pi/2 yields Schwarz's P surface; this entry is the only other angle for which the resulting associate surface - the gyroid - is embedded.

			0.66348297051143480805756884743723...
In degrees: 38.0147739891080681076130861019883...

A. Schoen, Infinite Periodic Minimal Surfaces Without Self-Intersections, NASA Technical Note D-5541, 1970.
A. Schoen, Triply-periodic minimal surfaces
Wikipedia, Associate family

Cf. A249282 (K(1/4)), A249283 (K(3/4)).

First@ RealDigits@ N[ArcTan[EllipticK[1/4] / EllipticK[3/4]], 120]

atan(ellK(1/2)/ellK(sqrt(3/4))) \\ Charles R Greathouse IV, Feb 05 2025

Equals arctan(K(1/4) / K(3/4)), where K is the complete elliptic integral of the first kind.

cp's OEIS Frontend

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

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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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A338004 Decimal expansion of the angle of association yielding the gyroid relative to Schwarz's D surface.

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