A249283 -id:A249283 - cp's OEIS frontend

A068521 Decimal expansion of agm(1, 2).

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

Offset: 1

Benoit Cloitre, Mar 21 2002

This is the arithmetic-geometric mean of 1 and 2, given by u(1) = 1, v(1) = 2, u(n+1) = (u(n)+v(n))/2, v(n+1) = sqrt(u(n)*v(n)); agm(1,2) = lim u(n) = lim v(n).

Schneider proved that this constant is transcendental. - Charles R Greathouse IV, Feb 03 2025

			1.45679103104690686918643238326508197497386394322130559079417238326792645458025...

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

Cf. A084895 (agm(1,3)), A084896 (agm(1,4)), A084897 (agm(1,5)), A000796, A249283.

evalf(GaussAGM(1, 2), 144);  # Alois P. Heinz, Jul 05 2023
evalf(Pi/EllipticK(sqrt(3)/2), 107); # or
evalf(3*Pi/(4*EllipticK(1/3)), 107); # Vaclav Kotesovec, Mar 28 2024

RealDigits[ ArithmeticGeometricMean[1, 2], 10, 107] // First (* Jean-François Alcover, Feb 06 2013 *)
RealDigits[N[3Pi/(4EllipticK[1/9]), 107]][[1]] (* Jean-François Alcover, Jun 02 2019 *)
RealDigits[N[Pi/EllipticK[3/4], 107]][[1]] (* or *)
RealDigits[N[Pi/(2*EllipticK[-3]), 107]][[1]] (* Vaclav Kotesovec, Mar 28 2024 *)

agm(1,2) \\ Charles R Greathouse IV, Mar 03 2016

Equals Pi/EllipticK(3/4) = A000796 / A249283. - Amiram Eldar, Apr 28 2025

A249282 Decimal expansion of K(1/4), where K is the complete elliptic integral of the first kind.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 24 2014

			1.685750354812596042871203657799076989500800894141089...

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)), A249283 (K(3/4)), A000796, A084895.

evalf(EllipticK(1/2), 120); # Vaclav Kotesovec, Apr 22 2015

RealDigits[EllipticK[1/4], 10, 102] // First

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

From Paul D. Hanna, Mar 25 2024: (Start)
K(1/4) = Pi/2 * Sum_{n>=0} binomial(2*n,n)^2/16^n * (1/4)^n.
K(1/4) = Pi/2 * sqrt( Sum_{n>=0} binomial(2*n,n)^3/16^n * (m*(1-m))^n ), where m = 1/4. (End)
Equals Pi/agm(1, 3) = A000796 / A084895. - Amiram Eldar, Apr 28 2025

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.

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A068521 Decimal expansion of agm(1, 2).

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A249282 Decimal expansion of K(1/4), where K is the complete elliptic integral of the first kind.

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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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Mathematica

PARI

Formula