y-cruncher-Formulas/Official Formulas/Log(2) - Zuniga (2024-i).cfg at master · Mysticial/y-cruncher-Formulas · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
//  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:    January 12, 2024
//
//    Value: 0.69314718055994530941723212145817656807550013436025...
//
//                                               _                    _
//               1   inf (  (1/221184)^n P(n)   |  1    1/2  1/3  2/3  |  )
//    Log(2) = ----- SUM ( -------------------- |                      |  )
//               3   n=1 (  n(2n-1)(3n-1)(3n-2) |_1/12 5/12 7/12 11/12_|n )
//
//    or
//
//               1   inf (         P(n)           k=n  (         k(2k-1)(3k-1)(3k-2)        ) )
//    Log(2) = ----- SUM ( ---------------------  PROD ( ---------------------------------- ) )
//               3   n=1 ( n(2n-1)(3n-1)(3n-2)    k=1  ( 192 (12k-1)(12k-5)(12k-7)(12k-11)  ) )
//
//
//    P(n)   =   686430*n^3 - 742257*n^2 + 223397*n - 13858
//
//                 _                  _
//                |  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 = GAMMA(w + n)/GAMMA(w) = w (w + 1) (w + 2) .... (w + n - 1)
//
//    Cost 1.3001
//
//    https://mathoverflow.net/questions/463436/are-these-fast-convergent-series-for-log2-log3-and-log5-already
//
//

{
    NameShort : "Log(2)"
    NameLong : "Log(2)"
    AlgorithmShort : "Zuniga (2024-i)"
    AlgorithmLong : "Zuniga (2024-i)"
    Formula : {
        SeriesHypergeometric : {
            CoefficientP : 1
            CoefficientQ : 0
            CoefficientD : 3
            PolynomialP : [-13858 223397 -742257 686430]
            PolynomialQ : [73920 -1271808 5253120 -7962624 3981312]
            PolynomialR : [0 -2 13 -27 18]
        }
    }
}