User blog:B1mb0w/Beta Function Code Version 3

Beta Function - Sequence Generating Code
The Beta Function has been defined using program code shown below.

A separate blog will be written to explain how Sequence Generator Code is compiled and executed using a normal programming language ... Work in Progress.

Sequence Generating Code Version 3
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Granularity Examples \(\beta(6.838,3)\) to \(\beta(9,3)\)
Version 3 makes it possible to access ordinals in the following range:

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Granularity Examples \(\beta(10.079,4)\) to \(\beta(16,4)\)
When we use base \(v = 4\) we generate more undesired values as in this example:

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Valid Sequence Counts
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Test Bed for Version 3
Below is the test bed and various results using version 3.

\(\beta(3.141,3) = f_{\omega + 1}(3)\)

\(\beta(3.4417,3) = f_{\omega.2}(3)\)

\(\beta(3.9485,3) = f_{\omega^2}(3)\)

\(\beta(4.53,3) = f_{\omega^2.2}(3)\)

\(\beta(5.1963,3) = f_{(\omega\uparrow\uparrow 2)}(3)\)

\(\beta(5.3777,3) = f_{(\omega\uparrow\uparrow 2).2}(3)\)

\(\beta(5.5655,3) = f_{(\omega\uparrow\uparrow 2).(\omega)}(3)\)

\(\beta(5.9612,3) = f_{(\omega\uparrow\uparrow 2)^2}(3)\)

\(\beta(6.1694,3) = f_{(\omega\uparrow\uparrow 2)^2.2}(3)\)

\(\beta(6.83855,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(3)\)

\(\beta(6.917229885,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}.2}(3)\)

\(\beta(7.324573,3) = f_{(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega)}(3)\)

\(\beta(7.84517,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(3)\)

\(\beta(8.5974,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}}(3)\)

\(\beta(9,3) = f_{\varphi(1,0)}(3)\)

\(\beta(6.34962,4) = f_{(\omega\uparrow\uparrow 2)}(4)\)

\(\beta(7.407,4) = f_{(\omega\uparrow\uparrow 2)^3}(4)\)

\(\beta(8,4) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(4)\)

\(\beta(8.314075,4) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(4)\)

\(\beta(8.979697,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2}}(4)\)

\(\beta(9.698609,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3}}(4)\)

\(\beta(9.887156,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega}}(4)\)

\(\beta(10.07937,4) = f_{(\omega\uparrow\uparrow 3)}(4)\)

\(\beta(11.75788,4) = f_{(\omega\uparrow\uparrow 3)^3}(4)\)

\(\beta(12.699209,4) = f_{(\omega\uparrow\uparrow 3)^{\omega}}(4)\)

\(\beta(14.254379491,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega}}}(4)\)

\(\beta(15.101989005,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2}}}(4)\)

\(\beta(15.69488145,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3}}}(4)\)

\(\beta(15.89764036,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.2}}}(4)\)

\(\beta(15.94873806,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3}}}(4)\)

\(\beta(15.9871691,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}}}(4)\)

\(\beta(15.995721886,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}.3}}(4)\)

\(\beta(15.9973264,4) = f_{1}^{6}(f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}.3 + 3}}(4))\)

\(\beta(15.99759342,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}.3 + 3}.(\omega)}(4)\)

\(\beta(16.0000001,4) = f_{\varphi(1,0)}(4)\)

WORK IN PROGRESS