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

Author

A.H.M. Smeets, Sep 13 2018

Keywords

• nonn
• cons

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).

Examples

			0.808894930127211...
		

Extensions

More terms from Jon E. Schoenfield, May 26 2019