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Note G and the calculation of Bernoulli numbers
See Bernoulli numbers in Wikipedia for details on Bernoulli numbers.
They may have been selected as an example because:
- They cannot be enumerated using the Difference Engine
- The values seem to be fairly random
- The values require a machine that can handle very large numbers
- In approximations of trigonometric functions, they allow less calculating steps than other approximations known at that time. Note that Tchebysheff's Approximation was published several years later (1859).
- They are a sequence where the n-th number can be calculated from the previous numbers
Note that variants for the indexing and sign for Bernoulli numbers were proposed since that time; the later forms allow shorter writing of the formulas.
In Note G, equation 8 shows the recursion formula used by AAL:
`0 = -1/2 * (2n-1)/(2n+1) + B_1 <sup>((1))</sup> /(2*3*4)) + ... + B_(2n-1)`
and in terms of the number to be calculated:
`-B_(2n-1) = -1/2 * (2n-1)/(2n+1) + B_1 ((2n)/2) + B_3 ((2n*(2n-1)*(2n-2)) /(2*3*4)) + B_5 ((2n*(2n-1)... (2n-4)) /(2*3*4*5*6)) +...`
Using `A_i` as abbreviation for the factors gives
`B_(2n-1) = A_0(n) + A_1(n) * B_1 + A_3(n) * B_3 + ... + A_5(n) * B_(2n-3)`
In the text, AAL wrotes: A1, A3&c. being … functions of n …, but does not use a corresponding notation in the equation, which was done in the above equation.
Thus, the coefficients – which are not given in Note G, are:
`A_0(n) = 1/2 * (2n-1)/(2n+1)`
`A_1(n) = -(2n) / 2`
`A_3(n) = A_1(n) * (2n-1)/3 * (2n-2)/4`
`A_5(n) = A_3(n) * (2n-3)/5 * (2n-4)/6`
This makes clear, that for `A_5` and the following the calculation steps are structurally equal and thus can be calculated within a loop.
Note the signs in contrast to Note G.
(to be continued)