A332093 -id:A332093 - cp's OEIS frontend

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)).

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

Offset: 1

M. F. Hasler, Sep 18 2020

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.

Other 3-argument generalizations of the AGM have been proposed (cf. A332093) which will give different values for AGM(1,1,2).

			1.294575108116612643448643498210035367403797272156424586808664172...

Vladimir Reshetnikov, Arithmetic-geometric mean of 3 numbers, math.StackExchange.com, May 22 2016.
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.

Cf. A332092 (AGM(1,2,2)), A332093 (AGM(1,2,3)).

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

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

A342813 Decimal expansion of the limit of AGM(1, 2, ..., n)/n.

Original entry on oeis.org

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

Offset: 0

Ben Whitmore, Mar 22 2021

The two-parameter arithmetic-geometric mean function AGM is defined by taking the limit of the sequence of iterates of the map (x, y) -> ((x+y)/2, sqrt(x*y)). This can be extended to an arbitrary finite sequence of numbers by defining AGM(x(1), ..., x(n)) = AGM((x(1)+...+x(n))/n, (x(1)*...*x(n))^(1/n)). Different extensions of the definition to more than two parameters are also possible, such as the one used in A332093.

			0.431407125466772950330229198641630937300926634224766278636544...

Cf. A332093.

RealDigits[Pi/4 * (1/2 + 1/E) / EllipticK[((E-2)/(E+2))^2], 10, 100][[1]]
RealDigits[ArithmeticGeometricMean[(2 + E)/(4 E), 1/Sqrt[2 E]], 10, 100][[1]] (* Jan Mangaldan, Dec 07 2021 *)

Pi/4 * (exp(-1)+.5) / ellK(1-4/(exp(1)+2)) \\ Charles R Greathouse IV, Feb 05 2025

agm(exp(-1)/2+1/4, exp(-1/2)/sqrt(2)) \\ Charles R Greathouse IV, Feb 05 2025

Equals Pi/4 * (1/2 + 1/e) / K(((e-2)/(e+2))^2) where K is the complete elliptic integral of the first kind.

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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)).

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A342813 Decimal expansion of the limit of AGM(1, 2, ..., n)/n.

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