User blog:B1mb0w/Valid Sequence Counts

Valid Sequence Counts
In my blog on Program Code Version 4 (using Sequence Generator code) there is a reference to the number of Valid Sequences that can be generated by the Sequence Generating code. This blog will estimate the number and the growth rate of these numbers.

This blog will illustrate how to approach calculating Alpha numbers and will partially explain the pseudo code algorithm for how it works.

It will be useful to refer to the blog on Sequence Generating Code for more background information.

Alpha Number for \(\zeta_0\)
The following Alpha Function is a good place to start. The sequence for the \(\zeta_0\) ordinal is shown here. The table calculates the Alpha Number required to generate this sequence.

\(\alpha(100.78626719) = J_8(<2,<0,0,<0,1>,<0,0>,0>,0,<0,0>,<0,0>,<0,0>>,2,3)\)

\(= f_{\varphi(2,0)}^2(3) = f_{\zeta_0}^2(3)\)

A final calculation is required with this sum total as follows: \(10^{2.003401361} = 100.7862672\) to arrive at the Alpha Number. Some sequence numbers are adjusted for fine tuning purposes. The rationale is that the numbers 2 and 3 are minimum values, therefore sequence numbers of 0 are sufficient to represent the minimums in each case.

Alpha Number for \(\zeta_1\)
Here is the calculation for another ordinal that uses similar calculations.

\(\alpha(100.8426651) = J_8(<2,<0,0,<0,1>,<0,1>,0>,0,<0,0>,<0,0>,<0,0>>,2,3)\)

\(= f_{\varphi(2,1)}^2(3) = f_{\zeta_1}^2(3)\)

The final calculation in this case is: \(10^{2.003644315} = 100.8426651\) to arrive at the Alpha Number.

Alpha Number for \(\zeta_0.2\)
A final calculation illustrates the range possible.

\(\alpha(100.7862763) = J_8(<2,<0,0,<0,1>,<0,0>,0>,0,<0,0>,<0,1>,<0,0>>,2,3)\)

\(= f_{\varphi(2,0).2}^2(3) = f_{\zeta_0.2}^2(3)\)

Final calculation of this Alpha Number is: \(10^{2.0034014} = 100.7862763\)