A319215 Decimal expansion of AGHM(1,i,1+i)/(1+i), where i is the imaginary unit and AGHM stands for arithmetic-geometric-harmonic mean of a triple of numbers.

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

Offset

0,1

Comments

As AGM(x1,x2) is the well-known arithmetic-geometric mean of a pair of numbers x1 and x2, we can also define the AGHM(x1,x2,x3) as the arithmetic-geometric-harmonic mean of a triple of numbers x1, x2 and x3.

These three means were chosen because the arithmetic mean is the power mean with power = 1, the geometric mean is the power mean with power = 0 (lim_{power -> 0}) and the harmonic mean is the power mean with power = -1.

Definition of AGHM(x1,x2,x3), for arbitrary triple x1,x2,x3:

x1(0) = x1, x2(0) = x2, x3(0) = x3,

x1(n) = (x1(n-1) + x2(n-1) + x3(n-1))/3,

x2(n) = (x1(n-1) * x2(n-1) * x3(n-1))^(1/3),

x3(n) = 3/(1/x1(n-1) + 1/x2(n-1) + 1/x3(n-1)),

lim_{n -> inf} x1(n) = lim_{n -> inf} x2(n) = lim_{n -> inf} x3(n) = AGHM(x1,x2,x3).

Links

Table of n, a(n) for n=0..86.

Example

0.808894930127211...

Crossrefs

Sequence in context: A021557 A242943 A358941 * A200513 A200523 A118540

Adjacent sequences: A319212 A319213 A319214 * A319216 A319217 A319218

Keyword

nonn,cons

Author

A.H.M. Smeets, Sep 13 2018

Extensions

More terms from Jon E. Schoenfield, May 26 2019

Status

approved