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

1,2

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).

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

Example

1.45679103104690686918643238326508197497386394322130559079417238326792645458025...

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 *)

Prog

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

Crossrefs

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

Sequence in context: A075341 A309358 A143789 * A196697 A068318 A347932

Adjacent sequences: A068518 A068519 A068520 * A068522 A068523 A068524

Keyword

easy,nonn,cons

Author

Benoit Cloitre, Mar 21 2002

Status

approved