A319215 - OEIS

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

(list; constant; graph; refs; listen; history; text; internal format)

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