

A332093


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


2



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

1,2


COMMENTS

The Arithmeticgeometric 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 kth roots of the normalized kth 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 both converge, and their limits (thus that of all components) must be the same.
Has this multivariable AGM already been studied somewhere? Any references in that sense or formulas are welcome.
Other 3argument generalizations of the AGM have been proposed, which all give different values whenever the three arguments are not all equal: replacing P(a,b,c) by (agm(a,b), agm(b,c), agm(a,c)) or (agm(a,agm(b,c)), cyclic...) one gets 1.9091574... resp. 1.9091504..., but these are less straightforwardly generalized to a symmetric function in more than 3 arguments. Using the average of the kth roots rather than the root of the average (normalized elementary symmetric polynomial) yields 1.89321.... See the two StackExchange links and discussion on the mathfun list. [Edited by M. F. Hasler, Sep 23 2020]


LINKS

Table of n, a(n) for n=1..101.
Brad Klee, Iterated averaging of triples, mathfun list (available for subscribers), Sep 18 2020.
User Mathlover, To find the limit of three terms mean iteration, math.StackExchange.com, Jul 12 2013.
Vladimir Reshetnikov, Arithmeticgeometric mean of 3 numbers, math.StackExchange.com, May 22 2016.
Wikipedia, Arithmeticgeometric mean, created Jan 2, 2002.
Wikipedia, Elementary symmetric polynomial, created Jan 28, 2005.


EXAMPLE

1.90992623354081532372267510978753355913562440802728405833885556866...


PROG

(PARI) f(k, x, S)={forvec(i=vector(k, i, [1, #x]), S+=vecprod(vecextract(x, i)), 2); S/binomial(#x, k)} \\ normalized kth 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, 3])\.1^100)


CROSSREFS

Cf. A332091 = AGM(1,1,2), A332092 = AGM(1,2,2).
Cf. other sequences related to the AGM (of two numbers): A061979, A080504, A090852 ff, A127758 ff.
Sequence in context: A242400 A197264 A273086 * A056965 A347688 A151949
Adjacent sequences: A332090 A332091 A332092 * A332094 A332095 A332096


KEYWORD

nonn,cons


AUTHOR

M. F. Hasler, Sep 18 2020


STATUS

approved



