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Lemniscate - Zuniga (2023-vi).cfg
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Lemniscate - Zuniga (2023-vi).cfg
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// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename".
//
// y-cruncher custom custom:"filename.cfg"
//
// Author: Jorge Zuniga
// Date: October 3, 2023
//
// Value: 5.24411510858423962092967917978223882736550990286324...
//
// _ _
// inf ( (1/4096)^n P(n) | 1/4 3/4 3/8 7/8 | )
// 2 omega = SUM (-------------------------- | | )
// n=1 ( (4n-1)(4n-3)(8n-1)(8n-5) |_ 1/16 5/16 9/16 13/16 _|n )
//
// Or
// inf ( P(n) k=n ( (4k-1)(4k-3)(8k-1)(8k-5) ) )
// 2 omega = SUM (-------------------------- PROD (-----------------------------------) )
// n=1 ( (4n-1)(4n-3)(8n-1)(8n-5) k=1 ( 64 (16k-3)(16k-7)(16k-11)(16k-15) ) )
//
//
// P(n) = 1397760*n^3 - 1844096*n^2 + 722528*n - 79872
//
// _ _
// | a b c ... z | (a)_n (b)_n (c)_n ... (z)_n
// | | = -------------------------------
// |_ A B C ... Z _|n (A)_n (B)_n (C)_n ... (Z)_n
//
//
// (w)_n Pochhammer Symbol. (w)_n = w (w + 1) (w + 2) .... (w + n - 1)
//
// Cost = 1.9236
//
{
NameShort : "Lemniscate"
NameLong : "Lemniscate Constant"
AlgorithmShort : "Zuniga (2023-vi)"
AlgorithmLong : "Zuniga (2023-vi)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
PolynomialP : [-79872 722528 -1844096 1397760]
PolynomialQ : [221760 -2248704 7307264 -9437184 4194304]
PolynomialR : [15 -224 1040 -1792 1024]
}
}
}