Alpha Function Code Version 8

Alpha Function Code Version 8[]

This version of The Alpha Function has been changed to fix some errors from the last version and to 're-calibrate' the function and make the range more interesting.

The function code is still based on The S Function (Version 2), with a growth rate of \(f_{svo}(n)\).

Changes in Version 8[]

Version 8 has been changed from Version 7 to fix some errors and to 're-calibrate' the function and make the range more interesting.

The errors in the last version were due to the simple code used to generate the T() functions. The S() functions were successfully generating to their maximum range. The T() functions were not. Code changes to \(t_x\) and \(u_x\) have been made to correct this.

The next change was to reduce the range of the input real number parameter to between 0 and 10,000. The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. The real number is manipulated by Sequence Generating Code (see below) to create a finite sequence of finite integers that represents a unique combination of S and T functions which can be translated into unique finite integers (up to \(f_{svo}(n)\) for any n).

The Alpha Function will still translate unique real numbers into any and every finite integer (up to \(f_{svo}(n)\) for any \(n\)).

Example of Changes in Version 8[]

Version 8 will generate string combinations of S and T functions. Each combination uniquely belongs to an ascending order of all sequences. Therefore each sequence can be assigned a finite ordinal value. The growth rate of these combinations is \(f_{svo}(n)\) for any \(n\)).

\(\alpha(0) = 0\)

\(\alpha(1.5) = 2\)

\(\alpha(1.75) = S(2,1,1) = 4\)

\(\alpha(1.875) = S(2,1,1) + 2 = 6\)

\(\alpha(1.9375) = S(2,1,1) + 3 = 7\)

\(\alpha(2) = S(2,T(0),1) = 8\)

\(\alpha(3) = S(S(2,T(0),1),1,1) = 16\)

\(\alpha(3.12) = S(S(2,T(0),1),1,1) + S(2,1,1) + 3 = 16 + 4 + 3 = 23\)

The growth rate can be seen to accelerate when we start introducing more complex T functions:

\(\alpha(4) = S(2,T(0) + 1,1) = f_{\omega+1}(2) = f_{\omega}(8)\)

\(\alpha(6) = S(S(2,S(T(0),1,1),1),1,1) + S(2,1,1) + 3 = S(S(2,S(T(0),1,1),1),1,1) + 4 + 3\)

\(\alpha(10) = S(2,T(1) + 1,1)\)

\(\alpha(20) = S(2,S(T(1),T(0),1),1) + 3\)

\(\alpha(50) = S(S(2,T(T(0) + 1) + T(0),1),1,1) + 1\)

\(\alpha(100) = S(2,T(T(1)) + 1,1)\)

\(\alpha(500) = S(2,T(T(1) + T(0)),1)\)

Comparing Alpha Function Values[]

From the above examples, it is interesting to compare:

\(\alpha(3.12) = \alpha(3) + \alpha(1.9375)\)

Here is a more complex example:

\(\alpha(6.76412776412775) = S(S(2,S(T(0),1,1),1),T(0) + S(S(2,T(0) + 1,1),T(0),S(S(8,4 + 3,4 + 3),4 + 2,S(S(8,4 + 3,4 + 2),4 + 2,S(S(8,4 + 3,4 + 1),4 + 2,S(8,4,1))))),1)\)

\(\alpha(6.78) = S(2,S(T(0),1,1) + 1,1)\)

\(\alpha(7.68550368550367) = S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),S(S(2,S(T(0),1,1),1),T(0) + S(S(2,T(0) + 1,1),T(0),S(S(8,4 + 3,4 + 3),4 + 2,S(S(8,4 + 3,4 + 2),4 + 2,S(S(8,4 + 3,4 + 1),4,1)))),1))\)

and

\(\alpha(7.68550368550367) = S(\alpha(6.78),S(T(0),1,1),\alpha(6.76412776412735))\)

Granularity Examples[]

These examples illustrate the fine detail in real numbers that can be used to access large numbers via the Alpha Function:

\(\alpha(5.85) = S(2,S(T(0),1,1),1)\)

\(\alpha(7.69) = S(2,S(T(0),1,1) + T(0),1)\)

\(\alpha(8.61) = S(2,S(T(0),1,1) + T(0) + 1,1)\)

\(\alpha(9.54) = S(2,T(1),1)\)

\(\alpha(10) = S(2,T(1) + 1,1)\)

\(\alpha(11) = S(\alpha(10),S(T(0),\alpha(9.54) + S(\alpha(7.69),S(T(0),1,1) + 3,S(S(S(S(2,S(T(0),1,1) + 1,1),T(0) + 1,1),T(0),3),6,S(\alpha(5.85),T(0) + 1,1))),1),1)\)

\(\alpha(11.3) = S(S(S(\alpha(10),T(1),7),3,S(S(2,T(0) + 1,1),T(0),1) + 1),1,\alpha(10) + S(\alpha(9.54),S(T(0),7,\alpha(5.85)),1))\)

\(\alpha(11.39) = S(\alpha(10),T(1),S(\alpha(9.54),S(S(S(T(0),5,4),4,2),1,S(S(T(0),3,1),1,S(S(2,T(0) + 1,1),T(0),1)) + 2),1))\)

\(\alpha(11.397) = S(\alpha(10),T(1),S(\alpha(9.54),S(S(T(0),S(\alpha(8.61),T(0) + 5,1) + S(\alpha(7.69),1,2) + 1,1),2,1) + T(0),1))\)

\(\alpha(11.3972) = S(\alpha(10),T(1),S(\alpha(9.54),S(T(0),S(S(S(\alpha(8.61),S(T(0),1,1) + T(0),3),T(0) + 7,1),T(0),1) + 1,S(\alpha(8.61),T(0) + 2,1)),1))\)

or

\(\alpha(11.3972) = … S(S(S(\alpha(8.61),S(T(0),1,1) + T(0),3),T(0) + 7,1),T(0),1) + 1,S(\alpha(8.61),T(0) + 2,1)),1))\)

\(\alpha(11.39723) = ... S(\alpha(8.61),S(T(0),1,1) + T(0),S(\alpha(7.69),1,1) + S(8,5,1) + S(S(8,3,1),2,7)),1),1))\)

\(\alpha(11.397234) = ... S(\alpha(8.61),S(T(0),1,1) + T(0),S(S(\alpha(7.69),S(T(0),1,1) + 4,1),4,S(\alpha(7.69),S(T(0),1,1) + 1,1))),1),1))\)

\(\alpha(11.3972345) = ... S(\alpha(8.61),S(T(0),1,1) + T(0),S(\alpha(7.69),S(T(0),1,1) + S(S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),16),S(S(2,T(0) + 1,1),T(0),16),1),1)),1),1))\)

or

\(\alpha(11.3972345) = ... ... S(S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),16),S(S(2,T(0) + 1,1),T(0),16),1),1)),1),1))\)

\(\alpha(11.397234504) = ... ... S(S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),S(\alpha(5.85),T(0) + 1,1) + \alpha(5.85)),T(0),1),1)),1),1))\)

\(\alpha(11.3972345049) = ... ... S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),S(\alpha(5.85),T(0) + S(S(2,T(0) + 1,1),T(0),1),4)),1)),1),1))\)

or

\(\alpha(11.3972345049) = ... ... ... S(S(2,T(0) + 1,1),T(0),1),4)),1)),1),1))\)

\(\alpha(11.39723450498) = ... ... ... S(S(2,T(0) + 1,1),T(0),S(S(8,6,3),4,1)),1)),1)),1),1))\)

\(\alpha(11.397234504981) = ... ... ... S(S(2,T(0) + 1,1),T(0),S(S(8,7,1),6,2)),1)),1)),1),1))\)

\(\alpha(11.3972345049816) = ... ... ... S(S(2,T(0) + 1,1),T(0),S(8,7,7)),1)),1)),1),1))\)

\(\alpha(11.3972345049817) = S(2,T(1) + T(0),1)\)

Growth Rate of the Alpha Function[]

The Alpha Function is now 're-calibrated' to accept real number inputs up to \(10,000\) at which point the Alpha Function will generate an S Function approaching:

\(\alpha(10000) = S(2,T^{\omega}(0),1) = \omega\)

In other words, the Alpha Function has been hard-coded to asymptotically reach infinity when \(r = 10000\).

Version 8 Code[]

The code for the Alpha Function uses Sequence Generator Code Syntax to generate sequences of finite integers.

The previous Version 7 code has been changed to set a variable in the first line of code \(V_v = 2\) instead of \(V_v = 3\), change the code for \(t_x\), and, add new code for \(u_x\) .

Version 8 code is now:

a_x = (V_v=2,n_x,[[n_x]])

n_x = (C_d<2,C_d>(0:(i_x,[[i_x]]),(V_n=1,s_0,[[s_0]])))

s_x = (x>(0:[[V_v]],[[s_x-1]]),

      t_x,t_x>(0:(n_c<V_n,n_c>(0:,[[]+[n_c]])),

            (n_C<V_n,V_n=C_d..n_C,[S([],[t_x],[n_C])],

                  s_x+1<t_x,[[s_x+1]])))

t_x = (C_t<2,C_t>(0:(n_x<V_n,[[n_x]]),(t_a,V_t=C_t..t_a,u_0,[[u_0]])))

u_x = (x>(0:[T([t_a])],[[u_x-1]]),

      t_b<V_t,t_b>(0:(t_c<V_t,t_c>(0:,[[]+[t_c]])),

            (t_C<V_t,V_t=C_t..t_C,[S([],[t_b],[t_C])],

                  u_x+1<t_b,[[u_x+1]])))

i_x = (C_x<V_v-U_x,[[C_x]+[U_x]])

Test Bed for Version 8[]

Below is the test bed and various results using version 8.

\(\alpha(0) = 0\)

\(\alpha(1) = 1\)

\(\alpha(1.5) = 2\)

\(\alpha(1.625) = 3\)

\(\alpha(1.75) = S(2,1,1) = 4\)

\(\alpha(1.8125) = S(2,1,1) + 1 = 4 + 1\)

Second Attempt

\(\alpha(0) = 0\)

\(\alpha(1) = 1\)

\(\alpha(1.5) = 2\)

\(\alpha(1.625) = 3\)

\(\alpha(1.75) = S(2,1,1) = 4\)

\(\alpha(1.8125) = S(2,1,1) + 1 = 4 + 1\)

\(\alpha(1.875) = S(2,1,1) + 2 = 4 + 2\)

\(\alpha(1.9375) = S(2,1,1) + 3 = 4 + 3\)

\(\alpha(2) = S(2,T(0),1) = 8\)

\(\alpha(2.25) = S(2,T(0),1) + 1 = 8 + 1\)

\(\alpha(2.5) = S(2,T(0),1) + 2 = 8 + 2\)

\(\alpha(2.625) = S(2,T(0),1) + 3 = 8 + 3\)

\(\alpha(2.75) = S(2,T(0),1) + S(2,1,1) = 8 + 4\)

\(\alpha(2.8125) = S(2,T(0),1) + S(2,1,1) + 1 = 8 + 4 + 1\)

\(\alpha(2.875) = S(2,T(0),1) + S(2,1,1) + 2 = 8 + 4 + 2\)

\(\alpha(2.9375) = S(2,T(0),1) + S(2,1,1) + 3 = 8 + 4 + 3\)

\(\alpha(3) = S(S(2,T(0),1),1,1) = 16\)

\(\alpha(3.0625) = S(S(2,T(0),1),1,1) + 1 = 16 + 1\)

\(\alpha(3.085) = S(S(2,T(0),1),1,1) + 2 = 16 + 2\)

\(\alpha(3.100625) = S(S(2,T(0),1),1,1) + 3 = 16 + 3\)

\(\alpha(3.1084375) = S(S(2,T(0),1),1,1) + S(2,1,1) = 16 + 4\)

\(\alpha(3.11234375) = S(S(2,T(0),1),1,1) + S(2,1,1) + 1 = 16 + 4 + 1\)

\(\alpha(3.11625) = S(S(2,T(0),1),1,1) + S(2,1,1) + 2 = 16 + 4 + 2\)

\(\alpha(3.12015625) = S(S(2,T(0),1),1,1) + S(2,1,1) + 3 = 16 + 4 + 3\)

\(\alpha(3.125) = S(S(2,T(0),1),1,1) + S(2,T(0),1) = 16 + 8\)

\(\alpha(4) = S(2,T(0) + 1,1) = 2048\)

\(\alpha(5) = S(S(2,T(0) + 1,1),T(0),1) + S(2,1,1) + 1 = S(2048,T(0),1) + 4 + 1\)

\(\alpha(6) = S(S(2,S(T(0),1,1),1),1,1) + S(2,1,1) + 3 = S(S(2,S(T(0),1,1),1),1,1) + 4 + 3\)

\(\alpha(7) = S(S(2,S(T(0),1,1) + 1,1),2,1) + S(S(2,T(0),1),1,1) + 1 = S(S(2,S(T(0),1,1) + 1,1),2,1) + 16 + 1\)

\(\alpha(8) = S(S(S(2,S(T(0),1,1) + T(0),1),S(2,1,1) + 1,S(S(2,T(0) + 1,1),T(0),1) + 1),2,S(S(2,S(T(0),1,1) + T(0),1),2,1) + S(S(2,S(T(0),1,1) + T(0),1),1,1) + S(S(S(2,T(0),1),S(2,1,1) + 2,1),1,2)) = S(S(S(2,S(T(0),1,1) + T(0),1),4 + 1,S(2048,T(0),1) + 1),2,S(S(2,S(T(0),1,1) + T(0),1),2,1) + S(S(2,S(T(0),1,1) + T(0),1),1,1) + S(S(8,4 + 2,1),1,2))\)

Third Attempt

\(\alpha(0) = 0\)

\(\alpha(1) = 1\)

\(\alpha(1.5) = 2\)

\(\alpha(1.625) = 3\)

\(\alpha(1.75) = S(2,1,1) = 4\)

\(\alpha(1.8125) = S(2,1,1) + 1 = 4 + 1\)

\(\alpha(1.875) = S(2,1,1) + 2 = 4 + 2\)

\(\alpha(1.9375) = S(2,1,1) + 3 = 4 + 3\)

\(\alpha(2) = S(2,T(0),1) = 8\)

\(\alpha(2.5) = S(2,T(0),1) + 1 = 8 + 1\)

\(\alpha(3) = S(2,T(0),1) + 2 = 8 + 2\)

\(\alpha(3.125) = S(2,T(0),1) + 3 = 8 + 3\)

\(\alpha(3.1875) = S(2,T(0),1) + S(2,1,1) = 8 + 4\)

\(\alpha(3.21875) = S(2,T(0),1) + S(2,1,1) + 1 = 8 + 4 + 1\)

\(\alpha(3.28125) = S(2,T(0),1) + S(2,1,1) + 2 = 8 + 4 + 2\)

\(\alpha(3.3125) = S(2,T(0),1) + S(2,1,1) + 3 = 8 + 4 + 3\)

\(\alpha(3.34375) = S(S(2,T(0),1),1,1) = 16\)

\(\alpha(3.46875) = S(S(2,T(0),1),1,1) + 1 = 16 + 1\)

\(\alpha(3.5) = S(S(2,T(0),1),1,1) + 2 = 16 + 2\)

\(\alpha(3.53125) = S(S(2,T(0),1),1,1) + 3 = 16 + 3\)

\(\alpha(3.546875) = S(S(2,T(0),1),1,1) + S(2,1,1) = 16 + 4\)

\(\alpha(3.5546875) = S(S(2,T(0),1),1,1) + S(2,1,1) + 1 = 16 + 4 + 1\)

\(\alpha(3.5625) = S(S(2,T(0),1),1,1) + S(2,1,1) + 2 = 16 + 4 + 2\)

\(\alpha(3.578125) = S(S(2,T(0),1),1,1) + S(2,1,1) + 3 = 16 + 4 + 3\)

\(\alpha(3.5859375) = S(S(2,T(0),1),1,1) + S(2,T(0),1) = 16 + 8\)

\(\alpha(4) = S(2,T(0) + 1,1) = 2048\)

\(\alpha(5) = S(S(2,T(0) + 1,1),T(0),1) + 1 = S(2048,T(0),1) + 1\)

\(\alpha(6) = S(2,S(T(0),1,1),1) + S(2,1,1) + 1 = S(2,S(T(0),1,1),1) + 4 + 1\)

\(\alpha(7) = S(S(2,S(T(0),1,1) + 1,1),1,1)\)

\(\alpha(8) = S(S(2,S(T(0),1,1) + T(0),1),1,1) + S(2,1,1) + 3 = S(S(2,S(T(0),1,1) + T(0),1),1,1) + 4 + 3\)

\(\alpha(9) = S(S(2,S(T(0),1,1) + T(0) + 1,1),1,S(2,1,1) + 2) + 3 = S(S(2,S(T(0),1,1) + T(0) + 1,1),1,4 + 2) + 3\)

\(\alpha(10) = S(2,T(1) + 1,1)\)

\(\alpha(15) = S(2,S(T(1),1,1) + 1,1)\)

\(\alpha(20) = S(2,S(T(1),T(0),1),1) + 1\)

\(\alpha(25) = S(2,S(T(1),T(0) + 1,S(T(0),1,1)),1) + S(S(2,S(T(1),T(0),1) + T(0) + 1,1),S(T(0),1,1) + T(0),1) + S(S(S(2,S(T(0),1,1) + 1,1),T(0) + 1,1),1,S(2,S(T(0),1,1),1))\)

\(\alpha(30) = S(S(2,S(S(S(S(T(1),S(T(0),1,1) + T(0),S(T(0),1,1) + T(0) + 1),S(T(0),1,1) + 1,1),T(0),S(T(0),1,1)),1,1),1),1,1) + 2\)

\(\alpha(31.6227766016838) = S(2,T(T(0)) + 1,1)\)

\(\alpha(35) = S(S(2,S(T(T(0)),1,1),1),1,1) + 1\)

\(\alpha(40) = S(S(2,S(T(T(0)),T(0),T(1) + 1) + S(T(1),1,1) + T(1) + 1,1),T(T(0)) + S(T(1),1,T(0) + 1) + S(T(0),1,1) + T(0) + 1,S(2,S(T(T(0)),T(0),T(1)) + T(0),1))\)

\(\alpha(45) = S(S(2,S(T(T(0)),S(T(1),1,1) + T(1),S(T(0),1,1)) + 1,1),1,S(S(2,T(0),1),1,1) + S(2,T(0),1) + 2) + S(2,1,1) + 1 = S(S(2,S(T(T(0)),S(T(1),1,1) + T(1),S(T(0),1,1)) + 1,1),1,16 + 8 + 2) + 4 + 1\)

\(\alpha(50) = S(2,T(T(0) + 1) + T(0),1) + 3\)

\(\alpha(55) = S(S(2,S(T(T(0) + 1),1,S(T(1),1,1) + 1),1),1,2) + S(2,S(S(S(S(T(1),S(T(0),1,1) + T(0),T(0) + 1),S(T(0),1,1) + 1,1),S(T(0),1,1),S(T(1),S(T(0),1,1) + T(0),1)),T(0),1),1)\)

\(\alpha(60) = S(2,S(S(T(T(0) + 1),S(T(1),T(0) + 1,1) + T(0),1),1,1) + 1,1) + S(2,1,1) + 1 = S(2,S(S(T(T(0) + 1),S(T(1),T(0) + 1,1) + T(0),1),1,1) + 1,1) + 4 + 1\)

\(\alpha(65) = S(2,S(T(T(0) + 1),S(T(T(0)),S(S(T(1),S(T(0),1,1) + T(0),1),1,S(T(1),T(0),1) + 1),T(1) + S(T(0),1,1)) + S(T(1),1,1) + 1,1) + T(T(0) + 1),1)\)

\(\alpha(70) = S(S(S(2,S(T(S(T(0),1,1)),S(T(0),1,1),1) + S(T(T(0) + 1),T(T(0)),1) + 1,1),T(S(T(0),1,1)) + 1,1),1,S(2,T(S(T(0),1,1)) + 1,1) + 2)\)

\(\alpha(75) = S(S(2,T(S(T(0),1,1) + 1) + S(T(0),1,1),1),1,2) + S(2,S(T(S(T(0),1,1)),S(T(T(0) + 1),1,1) + T(T(0)) + S(S(T(1),T(0) + 1,1),1,T(0) + 1) + T(0),1),1)\)

\(\alpha(80) = S(2,S(T(S(T(0),1,1) + 1),S(S(T(1),S(T(0),1,1),1),1,1) + T(1) + S(T(0),1,1) + 1,S(T(T(0)),1,T(1) + T(0)) + T(T(0)) + T(0)) + T(S(T(0),1,1) + 1),1)\)

\(\alpha(85) = S(S(S(2,S(T(S(T(0),1,1) + T(0)),1,1) + 1,1),T(0) + 1,1),1,S(2,S(S(T(S(T(0),1,1)),T(1) + 1,S(T(0),1,1) + 1),T(1),S(T(S(T(0),1,1)),1,1) + S(T(0),1,1)),1))\)

\(\alpha(90) = S(S(2,S(S(S(S(T(S(T(0),1,1) + T(0)),T(S(T(0),1,1)),1),T(0) + 1,1),T(0),1),1,S(S(S(T(1),S(T(0),1,1) + T(0) + 1,S(T(0),1,1)),S(T(0),1,1) + T(0),1),1,1) + 1),1),S(T(1),1,1),1)\)

\(\alpha(95) = S(2,S(T(S(T(0),1,1) + T(0) + 1),1,T(T(0) + 1)) + S(T(S(T(0),1,1) + 1),T(1) + T(0) + 1,S(T(1),T(0) + 1,1) + 1) + S(T(S(T(0),1,1)),S(T(1),S(T(0),1,1),1),1),1)\)

\(\alpha(100) = S(2,T(T(1)) + 1,1)\)

\(\alpha(200) = S(2,S(S(T(T(1)),T(0) + 1,T(0) + 1),1,1) + S(S(T(S(T(0),1,1) + T(0) + 1),S(T(1),S(T(0),1,1) + T(0),S(T(0),1,1) + 1) + 1,1),S(T(1),S(T(0),1,1),S(T(0),1,1) + T(0)),1),1)\)

\(\alpha(300) = S(S(2,T(T(1) + 1),1),T(0),3) + S(2,T(S(T(0),1,1)) + S(T(T(0)),1,1) + S(S(S(T(1),S(T(0),1,1),T(0) + 1),T(0),1),1,T(0)),1)\)

\(\alpha(400) = S(2,S(T(T(1) + 1),T(0),T(0) + 1),1)\)

Next Attempt on 1 Apr 2017

\(\alpha(0) = 0\)

\(\alpha(1) = 1\)

\(\alpha(1.5) = 2\)

\(\alpha(1.625) = 3\)

\(\alpha(1.75) = S(2,1,1) = 4\)

\(\alpha(1.8125) = S(2,1,1) + 1 = 4 + 1\)

\(\alpha(1.875) = S(2,1,1) + 2 = 4 + 2\)

\(\alpha(1.9375) = S(2,1,1) + 3 = 4 + 3\)

\(\alpha(2) = S(2,T(0),1) = 8\)

\(\alpha(2.25) = S(2,T(0),1) + 1 = 8 + 1\)

\(\alpha(2.5) = S(2,T(0),1) + 2 = 8 + 2\)

\(\alpha(2.625) = S(2,T(0),1) + 3 = 8 + 3\)

\(\alpha(2.75) = S(2,T(0),1) + S(2,1,1) = 8 + 4\)

\(\alpha(2.8125) = S(2,T(0),1) + S(2,1,1) + 1 = 8 + 4 + 1\)

\(\alpha(2.875) = S(2,T(0),1) + S(2,1,1) + 2 = 8 + 4 + 2\)

\(\alpha(2.94) = S(2,T(0),1) + S(2,1,1) + 3 = 8 + 4 + 3\)

\(\alpha(3) = S(S(2,T(0),1),1,1) = 16\)

\(\alpha(3.0625) = S(S(2,T(0),1),1,1) + 1 = 16 + 1\)

\(\alpha(3.09) = S(S(2,T(0),1),1,1) + 2 = 16 + 2\)

\(\alpha(3.1) = S(S(2,T(0),1),1,1) + 3 = 16 + 3\)

\(\alpha(3.105) = S(S(2,T(0),1),1,1) + S(2,1,1) = 16 + 4\)

\(\alpha(3.11) = S(S(2,T(0),1),1,1) + S(2,1,1) + 1 = 16 + 4 + 1\)

\(\alpha(3.115) = S(S(2,T(0),1),1,1) + S(2,1,1) + 2 = 16 + 4 + 2\)

\(\alpha(3.12) = S(S(2,T(0),1),1,1) + S(2,1,1) + 3 = 16 + 4 + 3\)

\(\alpha(3.125) = S(S(2,T(0),1),1,1) + S(2,T(0),1) = 16 + 8\)

\(\alpha(4) = S(2,T(0) + 1,1)\)

\(\alpha(5) = S(S(2,T(0) + 1,1),T(0),1) + S(2,1,1) + 1 = S(S(2,T(0) + 1,1),T(0),1) + 4 + 1\)

\(\alpha(6) = S(S(2,S(T(0),1,1),1),1,1) + S(2,1,1) + 3 = S(S(2,S(T(0),1,1),1),1,1) + 4 + 3\)

\(\alpha(7) = S(S(2,S(T(0),1,1) + 1,1),2,1) + S(S(2,T(0),1),1,1) + 1 = S(S(2,S(T(0),1,1) + 1,1),2,1) + 16 + 1\)

\(\alpha(8) = S(S(S(2,S(T(0),1,1) + T(0),1),S(2,1,1) + 1,S(S(2,T(0) + 1,1),T(0),1) + 1),2,S(S(2,S(T(0),1,1) + T(0),1),2,1) + S(S(2,S(T(0),1,1) + T(0),1),1,1) + S(S(S(2,T(0),1),S(2,1,1) + 2,1),1,2)) = S(S(S(2,S(T(0),1,1) + T(0),1),4 + 1,S(S(2,T(0) + 1,1),T(0),1) + 1),2,S(S(2,S(T(0),1,1) + T(0),1),2,1) + S(S(2,S(T(0),1,1) + T(0),1),1,1) + S(S(8,4 + 2,1),1,2))\)

\(\alpha(9) = S(S(S(2,S(T(0),1,1) + T(0) + 1,1),S(S(S(S(S(2,T(0) + 1,1),T(0),1),3,S(S(2,T(0),1),2,1)),2,S(2,1,1) + 2),1,1) + 1,S(2,T(0) + 1,1)),1,S(2,1,1) + 2) = S(S(S(2,S(T(0),1,1) + T(0) + 1,1),S(S(S(S(S(2,T(0) + 1,1),T(0),1),3,S(8,2,1)),2,4 + 2),1,1) + 1,S(2,T(0) + 1,1)),1,4 + 2)\)

\(\alpha(10) = S(2,T(1) + 1,1)\)

\(\alpha(15) = S(2,S(T(1),1,1) + 1,1) + 1\)

\(\alpha(20) = S(2,S(T(1),T(0),1),1) + 3\)

\(\alpha(25) = S(S(2,S(T(1),T(0) + 1,S(T(0),1,1)),1),1,S(S(2,T(1) + 1,1),1,S(2,1,1) + 2)) + 1 = S(S(2,S(T(1),T(0) + 1,S(T(0),1,1)),1),1,S(S(2,T(1) + 1,1),1,4 + 2)) + 1\)

\(\alpha(30) = S(S(2,S(S(S(S(T(1),S(T(0),1,1) + T(0),S(T(0),1,1) + T(0) + 1),S(T(0),1,1) + 1,1),T(0),S(T(0),1,1)),1,1),1),S(2,1,1),1) + 1 = S(S(2,S(S(S(S(T(1),S(T(0),1,1) + T(0),S(T(0),1,1) + T(0) + 1),S(T(0),1,1) + 1,1),T(0),S(T(0),1,1)),1,1),1),4,1) + 1\)

\(\alpha(31.6227766016838) = S(2,T(T(0)) + 1,1)\)

\(\alpha(35) = S(S(S(2,S(T(T(0)),1,1),1),3,1),2,S(2,1,1) + 1) = S(S(S(2,S(T(T(0)),1,1),1),3,1),2,4 + 1)\)

\(\alpha(40) = S(S(2,S(T(T(0)),T(0),T(1) + 1) + S(T(1),1,1) + T(1) + 1,1),S(T(T(0)),1,S(T(1),S(T(0),1,S(2,1,1) + 3) + S(T(0),1,S(2,1,1) + 2) + T(0) + 1,1)),1) = S(S(2,S(T(T(0)),T(0),T(1) + 1) + S(T(1),1,1) + T(1) + 1,1),S(T(T(0)),1,S(T(1),S(T(0),1,4 + 3) + S(T(0),1,4 + 2) + T(0) + 1,1)),1)\)

\(\alpha(45) = S(S(2,S(T(T(0)),S(T(1),1,1) + T(1),S(T(0),1,1)) + 1,1),S(S(2,T(T(0)),1),S(T(1),S(S(S(T(0),3,1),2,3),1,S(T(0),1,2)),1),1),1)\)

\(\alpha(50) = S(S(2,T(T(0) + 1) + T(0),1),1,1) + 1\)

\(\alpha(55) = S(S(2,S(T(T(0) + 1),1,S(T(1),1,1) + 1),1),S(S(S(2,T(0) + 1,1),T(0),1),S(S(2,T(0) + 1,1),S(2,1,1) + 1,S(S(S(2,T(0),1),3,S(2,1,1) + 3),2,2) + 1),1),1) = S(S(2,S(T(T(0) + 1),1,S(T(1),1,1) + 1),1),S(S(S(2,T(0) + 1,1),T(0),1),S(S(2,T(0) + 1,1),4 + 1,S(S(8,3,4 + 3),2,2) + 1),1),1)\)

\(\alpha(60) = S(S(2,S(S(T(T(0) + 1),S(T(1),T(0) + 1,1) + T(0),1),1,1) + 1,1),1,1) + S(2,1,1) + 3 = S(S(2,S(S(T(T(0) + 1),S(T(1),T(0) + 1,1) + T(0),1),1,1) + 1,1),1,1) + 4 + 3\)

\(\alpha(65) = S(2,S(T(T(0) + 1),S(T(T(0)),S(S(T(1),S(T(0),1,1) + T(0),1),1,S(T(1),T(0),1) + 1),T(1) + S(T(0),1,1)) + S(T(1),1,1) + 1,1) + T(T(0) + 1),1)\)

\(\alpha(70) = S(S(2,S(T(S(T(0),1,1)),S(T(0),1,1),1) + S(T(T(0) + 1),T(T(0)),1) + 1,1),T(S(T(0),1,1)) + S(T(1),S(T(0),S(2,S(T(T(0)),1,1) + 1,1),1),1),1)\)

\(\alpha(75) = S(S(2,T(S(T(0),1,1) + 1) + S(T(0),1,1),1),S(S(S(2,T(0) + 1,1),T(0),S(S(2,T(0),1),1,S(2,1,1) + 1) + 1),2,S(S(S(2,T(0) + 1,1),3,3),2,1)),1) = S(S(2,T(S(T(0),1,1) + 1) + S(T(0),1,1),1),S(S(S(2,T(0) + 1,1),T(0),S(8,1,4 + 1) + 1),2,S(S(S(2,T(0) + 1,1),3,3),2,1)),1)\)

\(\alpha(80) = S(2,S(T(S(T(0),1,1) + 1),S(S(T(1),S(T(0),1,1),1),1,1) + T(1) + S(T(0),1,1) + 1,S(T(T(0)),1,T(1) + T(0)) + T(T(0)) + T(0)) + T(S(T(0),1,1) + 1),1)\)

\(\alpha(85) = S(S(2,S(T(S(T(0),1,1) + T(0)),1,1) + 1,1),S(T(0),3,S(2,S(S(T(S(T(0),1,1)),T(1) + 1,S(T(0),1,1) + 1),T(1),S(T(S(T(0),1,1)),1,1) + S(T(0),1,1)),1)),1)\)

\(\alpha(90) = S(S(2,S(S(S(S(T(S(T(0),1,1) + T(0)),T(S(T(0),1,1)),1),T(0) + 1,1),T(0),1),1,S(S(S(T(1),S(T(0),1,1) + T(0) + 1,S(T(0),1,1)),S(T(0),1,1) + T(0),1),1,1) + 1),1),T(S(2,1,1)),1) = S(S(2,S(S(S(S(T(S(T(0),1,1) + T(0)),T(S(T(0),1,1)),1),T(0) + 1,1),T(0),1),1,S(S(S(T(1),S(T(0),1,1) + T(0) + 1,S(T(0),1,1)),S(T(0),1,1) + T(0),1),1,1) + 1),1),T(4),1)\)

\(\alpha(95) = S(2,S(T(S(T(0),1,1) + T(0) + 1),1,T(T(0) + 1)) + S(T(S(T(0),1,1) + 1),T(1) + T(0) + 1,S(T(1),T(0) + 1,1) + 1) + S(T(S(T(0),1,1)),S(T(1),S(T(0),1,1),1),1),1)\)

\(\alpha(100) = S(2,T(T(1)) + 1,1)\)

\(\alpha(200) = S(2,S(S(T(T(1)),T(0) + 1,T(0) + 1),1,1) + S(S(T(S(T(0),1,1) + T(0) + 1),S(T(1),S(T(0),1,1) + T(0),S(T(0),1,1) + 1) + 1,1),S(T(1),S(T(0),1,1),S(T(0),1,1) + T(0)),1),1)\)

\(\alpha(300) = S(S(S(S(2,T(T(1) + 1),1),S(T(0),1,1) + S(2,1,1) + 3,1),T(0),1),S(S(2,S(T(0),1,1) + 1,1),S(S(2,S(T(0),1,1),1),1,1) + S(S(2,T(0),1),S(2,1,1),1),1),1) = S(S(S(S(2,T(T(1) + 1),1),S(T(0),1,1) + 4 + 3,1),T(0),1),S(S(2,S(T(0),1,1) + 1,1),S(S(2,S(T(0),1,1),1),1,1) + S(8,4,1),1),1)\)

\(\alpha(400) = S(2,S(T(T(1) + 1),T(0),T(0) + 1),1) + 1\)

\(\alpha(500) = S(2,T(T(1) + T(0)),1)\)

\(\alpha(1000) = S(2,S(T(S(T(1),1,1)),S(T(S(T(0),1,1) + 1),S(T(S(T(0),1,1)),S(T(T(0) + 1),T(0) + 1,1) + S(S(T(T(0)),S(T(0),1,1) + T(0) + 1,S(S(T(1),S(T(0),1,1) + T(0),1),1,1)),S(T(0),1,1) + T(0),1),1),1),1),1)\)

\(\alpha(2000) = S(S(2,S(T(S(T(1),T(0),T(0) + 1)),1,S(S(S(T(T(0)),S(T(0),1,1) + 1,S(T(1),1,1) + 1),T(0),1),1,T(T(0)))) + 1,1),T(S(T(1),T(0),T(0))),1)\)

\(\alpha(3000) = S(S(2,S(S(T(S(T(1),S(T(0),1,1) + T(0),1) + S(T(1),1,T(0)) + 1),S(T(0),1,1) + T(0),1),1,T(0) + 1) + 1,1),S(T(T(0) + 2),S(2,S(T(1),S(T(0),1,1),1),1),1),1)\)

\(\alpha(4000) = S(2,S(T(S(T(T(0) + 1),1,S(S(S(T(T(0)),T(0) + 1,1),T(0),1),1,1) + T(0)) + 1),1,T(S(T(T(0)),1,1) + S(T(1),1,1)) + T(T(0))),1)\)

\(\alpha(5000) = S(2,T(S(S(T(S(T(0),1,1) + T(0) + 1),S(S(S(S(T(T(0)),S(T(0),1,1),1),T(0) + 1,1),T(0),S(T(1),T(0) + 1,1) + S(T(1),T(0),S(T(0),1,1) + T(0)) + 1),1,1),1),T(1),1)),1)\)

\(\alpha(6000) = S(2,T(S(T(S(T(1),T(0),T(0) + 1)),1,S(S(S(T(T(0)),S(T(0),1,1) + 1,S(T(1),1,1) + 1),T(0),1),1,T(T(0)))) + T(0)),1)\)

Next Attempt on 2 Apr 2017

\(\alpha(0) = 0\)

\(\alpha(1) = 1\)

\(\alpha(1.5) = 2\)

\(\alpha(1.625) = 3\)

\(\alpha(1.75) = S(2,1,1) = 4\)

\(\alpha(1.8125) = S(2,1,1) + 1 = 4 + 1\)

\(\alpha(1.875) = S(2,1,1) + 2 = 4 + 2\)

\(\alpha(1.9375) = S(2,1,1) + 3 = 4 + 3\)

\(\alpha(2) = S(2,T(0),1) = 8\)

\(\alpha(2.25) = S(2,T(0),1) + 1 = 8 + 1\)

\(\alpha(2.5) = S(2,T(0),1) + 2 = 8 + 2\)

\(\alpha(2.625) = S(2,T(0),1) + 3 = 8 + 3\)

\(\alpha(2.75) = S(2,T(0),1) + S(2,1,1) = 8 + 4\)

\(\alpha(2.8125) = S(2,T(0),1) + S(2,1,1) + 1 = 8 + 4 + 1\)

\(\alpha(2.875) = S(2,T(0),1) + S(2,1,1) + 2 = 8 + 4 + 2\)

\(\alpha(2.94) = S(2,T(0),1) + S(2,1,1) + 3 = 8 + 4 + 3\)

\(\alpha(3) = S(S(2,T(0),1),1,1) = 16\)

\(\alpha(3.0625) = S(S(2,T(0),1),1,1) + 1 = 16 + 1\)

\(\alpha(3.09) = S(S(2,T(0),1),1,1) + 2 = 16 + 2\)

\(\alpha(3.1) = S(S(2,T(0),1),1,1) + 3 = 16 + 3\)

\(\alpha(3.105) = S(S(2,T(0),1),1,1) + S(2,1,1) = 16 + 4\)

\(\alpha(3.11) = S(S(2,T(0),1),1,1) + S(2,1,1) + 1 = 16 + 4 + 1\)

\(\alpha(3.115) = S(S(2,T(0),1),1,1) + S(2,1,1) + 2 = 16 + 4 + 2\)

\(\alpha(3.12) = S(S(2,T(0),1),1,1) + S(2,1,1) + 3 = 16 + 4 + 3\)

\(\alpha(3.125) = S(S(2,T(0),1),1,1) + S(2,T(0),1) = 16 + 8\)

\(\alpha(3.34) = S(S(2,T(0),1),2,1) = 2048\)

\(\alpha(3.4076) = S(S(S(2,T(0),1),2,1),1,S(2,T(0),1) + 1) = S(2048,1,8 + 1)\)

\(\alpha(3.41600295) = S(S(S(2,T(0),1),2,1),1,S(S(2,T(0),1),1,S(2,1,1) + 1) + S(S(2,T(0),1),1,3) + 2) = S(2048,1,S(8,1,4 + 1) + 64 + 2)\)

\(\alpha(3.44) = S(S(2,T(0),1),2,3) = S(8,2,3)\)

\(\alpha(3.5) = S(S(2,T(0),1),3,1) = S(8,3,1)\)

\(\alpha(4) = S(2,T(0) + 1,1)\)

\(\alpha(5) = S(S(2,T(0) + 1,1),T(0),1) + S(2,1,1) + 1 = S(S(2,T(0) + 1,1),T(0),1) + 4 + 1\)

\(\alpha(5.7059715) = S(S(2,T(0) + 1,1),T(0),S(S(2,T(0),1),1,2) + S(S(2,T(0),1),1,1) + S(2,T(0),1) + S(2,1,1) + 3) = S(S(2,T(0) + 1,1),T(0),32 + 16 + 8 + 4 + 3)\)

\(\alpha(6) = S(S(2,S(T(0),1,1),1),1,1) + S(2,1,1) + 3 = S(S(2,S(T(0),1,1),1),1,1) + 4 + 3\)

\(\alpha(7) = S(S(2,S(T(0),1,1) + 1,1),2,1) + S(S(2,T(0),1),1,1) + 1 = S(S(2,S(T(0),1,1) + 1,1),2,1) + 16 + 1\)

\(\alpha(8) = S(S(S(2,S(T(0),1,1) + T(0),1),S(2,1,1) + 1,S(S(2,T(0) + 1,1),T(0),1) + 1),2,S(S(2,S(T(0),1,1) + T(0),1),2,1) + S(S(2,S(T(0),1,1) + T(0),1),1,1) + S(S(S(2,T(0),1),S(2,1,1) + 2,1),1,2)) = S(S(S(2,S(T(0),1,1) + T(0),1),4 + 1,S(S(2,T(0) + 1,1),T(0),1) + 1),2,S(S(2,S(T(0),1,1) + T(0),1),2,1) + S(S(2,S(T(0),1,1) + T(0),1),1,1) + S(S(8,4 + 2,1),1,2))\)

\(\alpha(9) = S(S(S(2,S(T(0),1,1) + T(0) + 1,1),S(S(S(S(S(2,T(0) + 1,1),T(0),1),3,S(S(2,T(0),1),2,1)),2,S(2,1,1) + 2),1,1) + 1,S(2,T(0) + 1,1)),1,S(2,1,1) + 2) = S(S(S(2,S(T(0),1,1) + T(0) + 1,1),S(S(S(S(S(2,T(0) + 1,1),T(0),1),3,2048),2,4 + 2),1,1) + 1,S(2,T(0) + 1,1)),1,4 + 2)\)

\(\alpha(9.54) = S(2,T(1),1)\)

\(\alpha(10) = S(2,T(1) + 1,1)\)

\(\alpha(15) = S(2,S(T(1),1,1) + 1,1) + 1\)

\(\alpha(20) = S(2,S(T(1),T(0),1),1) + 3\)

\(\alpha(25) = S(S(2,S(T(1),T(0) + 1,S(T(0),1,1)),1),1,S(S(2,T(1) + 1,1),1,S(2,1,1) + 2)) + 1 = S(S(2,S(T(1),T(0) + 1,S(T(0),1,1)),1),1,S(S(2,T(1) + 1,1),1,4 + 2)) + 1\)

\(\alpha(30) = S(S(2,S(S(S(S(T(1),S(T(0),1,1) + T(0),S(T(0),1,1) + T(0) + 1),S(T(0),1,1) + 1,1),T(0),S(T(0),1,1)),1,1),1),S(2,1,1),1) + 1 = S(S(2,S(S(S(S(T(1),S(T(0),1,1) + T(0),S(T(0),1,1) + T(0) + 1),S(T(0),1,1) + 1,1),T(0),S(T(0),1,1)),1,1),1),4,1) + 1\)

\(\alpha(31.44) = S(2,T(T(0)),1)\)