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

Author

Benoit Cloitre, Mar 21 2002

Keywords

• easy
• nonn
• cons

Comments

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

Examples

			1.45679103104690686918643238326508197497386394322130559079417238326792645458025...
		

Links

• 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.

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

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

Formula

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