

A332091


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


1



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

1,2


COMMENTS

See the main entry A332093 for more information on the multiargument AGM(...) used here. One main motivation for these entries is to find exact formulas for this function which seems not yet well studied in the literature, or at least for particular values like this one, A332092 = AGM(1,2,2) and A332093 = AGM(1,2,3). Any references to possibly existing works using this definition would be welcome.
Other 3argument generalizations of the AGM have been proposed (cf. A332093) which will give different values for AGM(1,1,2).


LINKS



EXAMPLE

1.294575108116612643448643498210035367403797272156424586808664172...


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, 1, 2])\.1^100)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



