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A332091 Decimal expansion of the arithmetic-geometric 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)).

%I A332091 #31 Dec 02 2020 08:15:36
%S A332091 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,
%T A332091 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,
%U A332091 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
%N A332091 Decimal expansion of the arithmetic-geometric 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)).
%C A332091 See the main entry A332093 for more information on the multi-argument 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.
%C A332091 Other 3-argument generalizations of the AGM have been proposed (cf. A332093) which will give different values for AGM(1,1,2).
%H A332091 Vladimir Reshetnikov, <a href="https://math.stackexchange.com/questions/1794795/arithmetic-geometric-mean-of-3-numbers">Arithmetic-geometric mean of 3 numbers</a>, math.StackExchange.com, May 22 2016.
%H A332091 User Mathlover, <a href="https://math.stackexchange.com/questions/442062/to-find-the-limit-of-three-terms-mean-iteration/3806368">To find the limit of three terms mean iteration</a>, math.StackExchange.com, Jul 12 2013.
%H A332091 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean">Arithmetic-geometric mean</a>, created Jan 2, 2002.
%e A332091 1.294575108116612643448643498210035367403797272156424586808664172...
%o A332091 (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
%o A332091 AGM(x)={until(x[1]<=x[#x],x=[sqrtn(f(k,x),k)|k<-[1..#x]]);vecsum(x)/#x}
%o A332091 default(realprecision,100);digits(AGM([1,1,2])\.1^100)
%Y A332091 Cf. A332092 (AGM(1,2,2)), A332093 (AGM(1,2,3)).
%Y A332091 Cf. other sequences related to the AGM (of two numbers): A061979, A080504, A090852 ff, A127758 ff.
%K A332091 nonn,cons
%O A332091 1,2
%A A332091 _M. F. Hasler_, Sep 18 2020