A332092 Decimal expansion of Arithmetic-geometric mean AGM(1, 2, 2) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 2, 2) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).

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

Offset: 1

Author

M. F. Hasler, Sep 18 2020

Keywords

• nonn
• cons

Comments

The Arithmetic-geometric mean of two values, AGM(x,y), is the limit of the sequence defined by iterations of (x,y) -> ((x+y)/2, sqrt(xy)). This can be generalized to any number of m variables by taking the vector of the k-th roots of the normalized k-th elementary symmetric polynomials in these variables, i.e., the average of all products of k among these m variables, with k = 1 .. m. After each iteration these m components are in strictly decreasing order unless they are all equal. Once they are in this order, the first one is strictly decreasing, the last one is strictly increasing, therefore they must all have the same limit.

Has this multi-variable AGM already been studied somewhere? Any contributions in that sense are welcome. (Other generalizations have also been proposed, cf. comments on StackExchange.)

Examples

			1.62885808884493884077629027798870804765763752833626903647603...
		

Links

• User Mathlover, To find the limit of three terms mean iteration, math.StackExchange.com, Jul 12 2013.
• Wikipedia, Arithmetic-geometric mean, created Jan 2, 2002.
• Wikipedia, Elementary symmetric polynomial, created Jan 28, 2005.

Crossrefs

Cf. other sequences related to the AGM (of two numbers): A061979, A080504, A090852 ff, A127758 ff.

Programs

PARI

f(k,x,S)={forvec(i=vector(k,i,[1,#x]), S+=vecprod(vecextract(x,i)),2); S/binomial(#x,k)} \\ normalized k-th elementary symmetric polynomial in x
AGM(x)={until(x[1]<=x[#x],x=[sqrtn(f(k,x),k)|k<-[1..#x]]);vecsum(x)/#x}
default(realprecision,100);digits(AGM([1,2,2])\.1^100)