User blog:B1mb0w/Alpha Spreadsheet q3.0

Alpha function code (Spreadsheet) version 3.0
Javascript code for my Alpha Function based on my Quantum Function is available in Google Sheets.

Javascript code in Google Sheets
You are free to use and copy the code as you like. Version 2.0 of the Javascript code in Google Sheets is available here:

Alpha Function Javascript

DRAFT Version 3.0 Examples
I have started working on a new version. Here is some work in progress:

\(\alpha(5) = Q^{2}(Q^{2}(1,Q(2,3)_*))\)

\(\alpha(6) = Q^{2}(Q(1,Q^{2}(2,3_*)))\)

\(\alpha(7) = Q^{2}(Q(1,Q^{4}(2,3_*)))\)

\(\alpha(8) = Q(Q^{15}(2,3_*))\)

\(\alpha(9) = Q(Q^{t_0(0)}(Q(1,t_0(0))),3)\)

\(\alpha(10) = Q(Q^{2}(Q^{t_0(0)}(t_0(0))_*,t_0(0)),3)\)

\(\alpha(11) = Q(Q(Q(Q^{2}(t_0(0)_*,t_0(0)),t_0(1)),t_0(1)),3)\)

\(\alpha(12) = Q(Q^{Q(Q(2,t_0(0)),t_0(0))}(t_0(1)_*,t_0(1)),3)\)

\(\alpha(13) = Q(Q(Q^{t_0(0)}(1,t_0(2)_*),t_0(2)),3)\)

\(\alpha(14) = Q(Q^{Q(2,t_0(0))}(t_0(2)_*,t_0(2)),3)\)

\(\alpha(15) = Q(Q^{Q^{t_0(0)}(t_0(1)_*,t_0(1))}(t_0(2)_*,t_0(2)),3)\)

\(\alpha(16) = Q(Q^{2}(Q^{t_0^{2}(0)}(2,t_0^{2}(0)_*)_*,t_0^{2}(0)),3)\)

\(\alpha(17) = Q(Q^{Q^{t_0(0)}(t_0(2)_*,t_0(2))}(t_0^{2}(0)_*,t_0^{2}(0)),3)\)

\(\alpha(18) = Q(t_0^{2}(Q(Q^{t_0(0)}(t_0^{[1]}(0)),t_0^{[1]}(0))),3)\)

\(\alpha(19) = Q(t_0^{2}(Q^{2}(Q(2,t_0^{[1]}(0))_*,t_0^{[1]}(0))),3)\)

\(\alpha(20) = Q(Q(Q(t_0^{2}(0),t_0^{2}(1)),t_0^{2}(1)),3)\)

\(\alpha(21) = Q(Q^{Q^{2}(t_0(2)_*,t_0(2))}(t_0^{2}(1)_*,t_0^{2}(1)),3)\)

\(\alpha(22) = Q(t_0^{2}(Q^{t_0(0)}(t_0^{[1]}(1))),3)\)

\(\alpha(23) = Q(t_0^{2}(Q(Q^{t_0(0)}(t_0^{[1]}(1)),t_0^{[1]}(1))),3)\)

\(\alpha(24) = Q(t_0^{2}(Q^{2}(Q^{2}(t_0^{[1]}(1))_*,t_0^{[1]}(1))),3)\)

\(\alpha(25) = Q^{3}(t_0^{2}(2),3_*)\)

\(\alpha(26) = Q(Q(Q(t_0(2),t_0^{2}(2)),t_0^{2}(2)),3)\)

\(\alpha(27) = Q(Q^{Q(t_0(0),t_0(0))}(t_0^{2}(2)_*,t_0^{2}(2)),3)\)

\(\alpha(28) = Q(Q^{Q^{t_0(0)}(t_0^{2}(0)_*,t_0^{2}(0))}(t_0^{2}(2)_*,t_0^{2}(2)),3)\)

\(\alpha(29) = Q(t_0^{2}(Q^{2}(t_0^{[1]}(2))),3)\)

\(\alpha(30) = Q(t_0^{2}(Q^{2}(2,t_0^{[1]}(2)_*)),3)\)

\(\alpha(31) = Q(t_0^{2}(Q(Q(1,t_0^{[1]}(2)),t_0^{[1]}(2))),3)\)

\(\alpha(32) = Q(t_0^{2}(Q(t_0(0),Q^{2}(t_0^{[1]}(2)_*,t_0^{[1]}(2)))),3)\)

\(\alpha(33) = Q(t_0^{2}(Q^{2}(Q^{t_0(0)}(1,t_0^{[1]}(2)_*)_*,t_0^{[1]}(2))),3)\)

\(\alpha(34) = Q(t_0(t_0(1,1)),3)\)

\(\alpha(35) = Q(t_0(t_0^{2}(1,1_*)),3)\)

\(\alpha(36) = Q(t_0^{Q(t_0(0),t_0(0))}(1,0_*),3)\)

\(\alpha(37) = Q(t_0^{t_0^{2}(0)}(1,0_*),3)\)

\(\alpha(38) = Q(t_0^{t_0^{2}(Q(t_0^{[1]}(2)))}(1,0_*),3)\)

\(\alpha(39) = Q(t_0(2,t_0^{2}(1)),3)\)

\(\alpha(40) = Q(t_0^{2}(2,Q(t_0^{[1]}(2,2))_*),3)\)

\(\alpha(41) = Q(t_0^{Q(t_0(1),t_0(1))}(2,0_*),3)\)

\(\alpha(42) = Q(t_0^{t_0^{2}(Q(t_0^{[1]}(2)))}(2,0_*),3)\)

\(\alpha(43) = Q(t_0^{t_0(1,0)}(2,0_*),3)\)

\(\alpha(44) = Q(t_0(Q(2,t_0(0)),0),3)\)

\(\alpha(45) = Q(t_0(Q^{2}(Q^{t_0(0)}(t_0(0))_*,t_0(0)),0),3)\)

\(\alpha(46) = Q(t_0(Q(Q^{2}(1,t_0(1)_*),t_0(1)),0),3)\)

\(\alpha(47) = Q(t_0(Q^{t_0(0)}(t_0(1)_*,t_0(1)),0),3)\)

\(\alpha(48) = Q(t_0(Q(t_0(0),t_0(2)),0),3)\)

\(\alpha(49) = Q(t_0(Q^{t_0(0)}(t_0(2)_*,t_0(2)),0),3)\)

\(\alpha(51) = Q(t_0(Q^{t_0(0)}(t_0^{2}(0)_*,t_0^{2}(0)),0),3)\)

\(\alpha(52) = Q(t_0(t_0^{2}(Q(t_0^{[1]}(0),t_0^{[1]}(0))),0),3)\)

\(\alpha(53) = Q(t_0(Q^{2}(Q^{t_0(0)}(t_0^{2}(1))_*,t_0^{2}(1)),0),3)\)

\(\alpha(54) = Q(t_0(t_0^{2}(Q(t_0^{[1]}(1))),0),3)\)

\(\alpha(55) = Q(t_0(t_0^{2}(Q^{2}(t_0^{[1]}(1)_*,t_0^{[1]}(1))),0),3)\)

\(\alpha(56) = Q(t_0(Q^{t_0(0)}(t_0^{2}(2)_*,t_0^{2}(2)),0),3)\)

\(\alpha(57) = Q(t_0(t_0^{2}(Q(t_0^{[1]}(2))),0),3)\)

\(\alpha(58) = Q(t_0(t_0^{2}(Q^{2}(t_0^{[1]}(2)_*,t_0^{[1]}(2))),0),3)\)

\(\alpha(59) = Q(t_0^{2}(Q(t_0^{[1]}(1,0))_*,0),3)\)

\(\alpha(60) = Q(t_0(t_0^{2}(1_*,1)),3)\)

\(\alpha(61) = Q(t_0^{2}(t_0^{2}(t_0^{[1]}(1,1))_*,1),3)\)

\(\alpha(62) = Q(t_0^{2}(Q(t_0^{[1]}(1,2))_*,2),3)\)

\(\alpha(63) = Q(t_0^{2}(1_*,Q(t_0(0),t_0(0))),3)\)

\(\alpha(64) = Q(t_0^{2}(1_*,Q^{t_0(0)}(t_0(2)_*,t_0(2))),3)\)

\(\alpha(65) = Q(t_0^{2}(1_*,t_0^{2}(2)),3)\)

\(\alpha(66) = Q(t_0^{2}(1_*,t_0(2,0)),3)\)

\(\alpha(67) = Q(t_0^{2}(1_*,t_0(t_0(0),0)),3)\)

\(\alpha(68) = Q(t_0^{2}(1_*,t_0(t_0(1,0),0)),3)\)

\(\alpha(69) = Q(t_0^{2}(Q(t_0^{[1]}(2,0))_*,0),3)\)

\(\alpha(70) = Q(t_0^{2}(2_*,1),3)\)

\(\alpha(71) = Q(t_0^{2}(Q(t_0^{[1]}(2,1))_*,1),3)\)

\(\alpha(72) = Q(Q(t_0^{2}(2_*,2),t_0^{2}(2_*,2)),3)\)

\(\alpha(73) = Q(t_0^{2}(Q(t_0^{[1]}(2,2))_*,2),3)\)

\(\alpha(74) = Q(t_0^{2}(2_*,t_0(0)),3)\)

\(\alpha(75) = Q(t_0^{2}(2_*,t_0(2)),3)\)

\(\alpha(76) = Q(t_0^{2}(2_*,t_0^{2}(Q(t_0^{[1]}(0)))),3)\)

\(\alpha(77) = Q(t_0^{2}(2_*,t_0^{2}(Q(t_0^{[1]}(2)))),3)\)

\(\alpha(78) = Q(t_0^{2}(2_*,t_0(2,0)),3)\)

\(\alpha(94.5) = Q(t_0^{2}(Q^{Q(Q(2,t_0(0)),t_0(0))}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(94.6) = Q(t_0^{2}(Q^{Q^{2}(t_0(0)_*,t_0(0))}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(94.8) = Q(t_0^{2}(Q^{Q^{2}(Q(1,t_0(0))_*,t_0(0))}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(95.0) = Q(t_0^{2}(Q^{Q^{2}(Q(2,t_0(0))_*,t_0(0))}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(95.1) = Q(t_0^{2}(Q^{t_0(1)}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(95.2) = Q(t_0^{2}(Q^{Q(t_0(0),t_0(1))}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(95.4) = Q(t_0^{2}(Q^{Q(Q(1,t_0(1)),t_0(1))}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(95.5) = Q(t_0^{2}(Q^{Q(Q(t_0(0),t_0(1)),t_0(1))}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(95.6) = Q(t_0^{2}(Q^{Q^{2}(t_0(1)_*,t_0(1))}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(95.7) = Q(t_0^{2}(Q^{Q^{2}(Q(t_0(0),t_0(1))_*,t_0(1))}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(95.9) = Q(t_0^{2}(Q^{Q^{t_0(0)}(t_0(1)_*,t_0(1))}(t_0(2)_*,t_0(2))_*,0),3)\)

\(\alpha(96.6) = Q(t_0^{2}(Q^{2}(t_0^{2}(0))_*,t_0(0)),3)\)

\(\alpha(96.7) = Q(t_0^{2}(Q^{t_0(0)}(1,t_0^{2}(0)_*)_*,0),3)\)

\(\alpha(96.8) = Q(t_0^{2}(Q^{t_0^{2}(0)}(2,t_0^{2}(0)_*)_*,0),3)\)

\(\alpha(96.9) = Q(t_0^{2}(Q(t_0(0),t_0^{2}(0))_*,0),3)\)

\(\alpha(97.0) = Q(t_0^{2}(Q(t_0(2),t_0^{2}(0))_*,0),3)\)

\(\alpha(97.1) = Q(t_0^{2}(Q^{t_0(0)}(Q(Q^{2}(t_0^{2}(0)),t_0^{2}(0)))_*,0),3)\)

\(\alpha(97.2) = Q(t_0^{2}(Q(Q^{t_0(0)}(1,t_0^{2}(0)_*),t_0^{2}(0))_*,0),3)\)

\(\alpha(97.3) = Q(t_0^{2}(Q(Q^{t_0(0)}(2,t_0^{2}(0)_*),t_0^{2}(0))_*,0),3)\)

\(\alpha(97.4) = Q(t_0^{2}(Q(Q(t_0(0),t_0^{2}(0)),t_0^{2}(0))_*,0),3)\)

\(\alpha(97.5) = Q(t_0^{2}(Q(Q(t_0(2),t_0^{2}(0)),t_0^{2}(0))_*,0),3)\)

\(\alpha(97.6) = Q(t_0^{2}(Q^{2}(Q(Q^{[1]}(Q(t_0^{2}(0)),t_0^{2}(0)))_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(97.7) = Q(t_0^{2}(Q^{2}(Q^{t_0(0)}(Q^{2}(1,t_0^{2}(0)_*))_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(97.8) = Q(t_0^{2}(Q^{2}(Q^{t_0(0)}(2,t_0^{2}(0)_*)_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(97.9) = Q(t_0^{2}(Q^{2}(Q(t_0(0),t_0^{2}(0))_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(98.0) = Q(t_0^{2}(Q^{2}(Q(Q^{t_0(0)}(t_0(1)_*,t_0(1)),t_0^{2}(0))_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(98.1) = Q(t_0^{2}(Q^{t_0(0)}(t_0^{2}(0)_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(98.3) = Q(t_0^{2}(Q^{Q(t_0(0),t_0(0))}(t_0^{2}(0)_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(98.5) = Q(t_0^{2}(Q^{Q^{2}(t_0(0)_*,t_0(0))}(t_0^{2}(0)_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(98.7) = Q(t_0^{2}(Q^{Q(t_0(1),t_0(1))}(t_0^{2}(0)_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(98.8) = Q(t_0^{2}(Q^{Q^{2}(t_0(1)_*,t_0(1))}(t_0^{2}(0)_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(98.9) = Q(t_0^{2}(Q^{Q^{t_0(0)}(t_0(1)_*,t_0(1))}(t_0^{2}(0)_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(99.2) = Q(t_0^{2}(Q^{Q(t_0(0),t_0(2))}(t_0^{2}(0)_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(99.3) = Q(t_0^{2}(Q^{Q(Q(t_0(0),t_0(2)),t_0(2))}(t_0^{2}(0)_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(99.4) = Q(t_0^{2}(Q^{Q^{t_0(0)}(t_0(2)_*,t_0(2))}(t_0^{2}(0)_*,t_0^{2}(0))_*,0),3)\)

\(\alpha(99.7) = Q(t_0^{2}(t_0^{2}(Q(t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(99.8) = Q(t_0^{2}(t_0^{2}(Q^{2}(t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(99.9) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(100.1) = Q(t_0^{2}(t_0^{2}(Q^{2}(1,t_0^{[1]}(0)_*))_*,0),3)\)

\(\alpha(100.2) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(1,t_0^{[1]}(0)_*))_*,0),3)\)

\(\alpha(100.4) = Q(t_0^{2}(t_0^{2}(Q(1,Q(2,t_0^{[1]}(0))))_*,0),3)\)

\(\alpha(100.5) = Q(t_0^{2}(t_0^{2}(Q(1,Q^{2}(2,t_0^{[1]}(0)_*)))_*,0),3)\)

\(\alpha(100.6) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(2,t_0^{[1]}(0)_*))_*,0),3)\)

\(\alpha(100.8) = Q(t_0^{2}(t_0^{2}(Q(t_0(0),Q(Q(t_0^{[1]}(0)),t_0^{[1]}(0))))_*,0),3)\)

\(\alpha(100.9) = Q(t_0^{2}(t_0^{2}(Q(Q^{t_0(0)}(t_0^{[1]}(0)),t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(101.1) = Q(t_0^{2}(t_0^{2}(Q(Q(1,t_0^{[1]}(0)),t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(101.2) = Q(t_0^{2}(t_0^{2}(Q(Q^{2}(1,t_0^{[1]}(0)_*),t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(101.3) = Q(t_0^{2}(t_0^{2}(Q(Q^{t_0(0)}(1,t_0^{[1]}(0)_*),t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(101.5) = Q(t_0^{2}(t_0^{2}(Q(Q(1,Q(2,t_0^{[1]}(0))),t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(101.6) = Q(t_0^{2}(t_0^{2}(Q(Q(1,Q^{2}(2,t_0^{[1]}(0)_*)),t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(101.7) = Q(t_0^{2}(t_0^{2}(Q(Q^{t_0(0)}(2,t_0^{[1]}(0)_*),t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(101.8) = Q(t_0^{2}(t_0^{2}(Q(t_0(0),Q^{2}(t_0^{[1]}(0)_*,t_0^{[1]}(0))))_*,0),3)\)

\(\alpha(101.9) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q(Q^{[1]}(Q(t_0^{[1]}(0)),t_0^{[1]}(0)))_*,t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(102.0) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q^{t_0(0)}(t_0^{[1]}(0))_*,t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(102.2) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q(1,t_0^{[1]}(0))_*,t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(102.3) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q^{2}(1,t_0^{[1]}(0)_*)_*,t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(102.4) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q^{t_0(0)}(1,t_0^{[1]}(0)_*)_*,t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(102.6) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q^{2}(2,t_0^{[1]}(0)_*)_*,t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(102.7) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q^{t_0(0)}(2,t_0^{[1]}(0)_*)_*,t_0^{[1]}(0)))_*,0),3)\)

\(\alpha(102.9) = Q(t_0^{2}(Q(t_0^{2}(1))_*,t_0(0)),3)\)

\(\alpha(103.0) = Q(t_0^{2}(Q^{t_0(0)}(Q^{2}(1,t_0^{2}(1)_*))_*,0),3)\)

\(\alpha(103.1) = Q(t_0^{2}(Q^{t_0(0)}(1,Q^{2}(2,t_0^{2}(1)_*)_*)_*,0),3)\)

\(\alpha(103.2) = Q(t_0^{2}(Q(t_0(1),t_0^{2}(1))_*,0),3)\)

\(\alpha(103.3) = Q(t_0^{2}(Q(t_0^{2}(0),t_0^{2}(1))_*,0),3)\)

\(\alpha(103.4) = Q(t_0^{2}(Q(t_0^{2}(Q(t_0^{[1]}(0))),t_0^{2}(1))_*,0),3)\)

\(\alpha(103.5) = Q(t_0^{2}(Q(Q^{t_0^{2}(0)}(t_0^{2}(1)),t_0^{2}(1))_*,0),3)\)

\(\alpha(103.6) = Q(t_0^{2}(Q(Q^{t_0^{2}(0)}(1,t_0^{2}(1)_*),t_0^{2}(1))_*,0),3)\)

\(\alpha(103.7) = Q(t_0^{2}(Q(Q^{t_0^{2}(0)}(2,t_0^{2}(1)_*),t_0^{2}(1))_*,0),3)\)

\(\alpha(103.8) = Q(t_0^{2}(Q(Q(t_0(2),t_0^{2}(1)),t_0^{2}(1))_*,0),3)\)

\(\alpha(103.9) = Q(t_0^{2}(Q(Q(t_0^{2}(0),t_0^{2}(1)),t_0^{2}(1))_*,0),3)\)

\(\alpha(104.0) = Q(t_0^{2}(Q(t_0(0),Q^{2}(Q(t_0^{2}(1))_*,t_0^{2}(1)))_*,0),3)\)

\(\alpha(104.1) = Q(t_0^{2}(Q^{2}(Q^{2}(1,t_0^{2}(1)_*)_*,t_0^{2}(1))_*,0),3)\)

\(\alpha(104.2) = Q(t_0^{2}(Q^{2}(Q^{t_0(0)}(Q^{2}(2,t_0^{2}(1)_*))_*,t_0^{2}(1))_*,0),3)\)

\(\alpha(106.34) = Q(t_0^{2}(Q(1,t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.35) = Q(t_0^{2}(Q(Q^{t_0(0)}(2,t_0^{2}(Q^{2}(t_0^{[1]}(1)))_*),t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.36) = Q(t_0^{2}(Q^{2}(Q(t_0(0),t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.37) = Q(t_0^{2}(Q^{t_0(0)}(t_0^{2}(Q^{2}(t_0^{[1]}(1)))_*,t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.38) = Q(t_0^{2}(Q^{t_0^{2}(0)}(t_0^{2}(Q^{2}(t_0^{[1]}(1)))_*,t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.39) = Q(t_0^{2}(Q(2,t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.40) = Q(t_0^{2}(Q(Q(t_0(0),t_0^{2}(Q^{2}(t_0^{[1]}(1)))),t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.41) = Q(t_0^{2}(Q^{2}(Q(t_0(0),t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.42) = Q(t_0^{2}(Q^{t_0(0)}(t_0^{2}(Q^{2}(t_0^{[1]}(1)))_*,t_0^{2}(Q^{2}(t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.44) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(106.63) = Q(t_0^{2}(Q^{2}(t_0^{2}(Q(1,t_0^{[1]}(1)))_*,t_0^{2}(Q(1,t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.64) = Q(t_0^{2}(t_0^{2}(Q(1,t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(106.65) = Q(t_0^{2}(Q^{t_0(0)}(t_0^{2}(Q(Q(1,t_0^{[1]}(1))))_*,t_0^{2}(Q(Q(1,t_0^{[1]}(1)))))_*,0),3)\)

\(\alpha(106.66) = Q(t_0^{2}(Q(t_0^{2}(Q^{2}(Q(1,t_0^{[1]}(1)))),t_0^{2}(Q^{2}(Q(1,t_0^{[1]}(1)))))_*,0),3)\)

\(\alpha(106.67) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q(1,t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.68) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q(1,t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(106.73) = Q(t_0^{2}(Q^{t_0(0)}(t_0^{2}(Q^{2}(1,t_0^{[1]}(1)_*))_*,t_0^{2}(Q^{2}(1,t_0^{[1]}(1)_*)))_*,0),3)\)

\(\alpha(106.74) = Q(t_0^{2}(Q(t_0^{2}(Q(Q^{2}(1,t_0^{[1]}(1)_*))),t_0^{2}(Q(Q^{2}(1,t_0^{[1]}(1)_*))))_*,0),3)\)

\(\alpha(106.75) = Q(t_0^{2}(t_0^{2}(Q(Q^{2}(1,t_0^{[1]}(1)_*)))_*,0),3)\)

\(\alpha(106.76) = Q(t_0^{2}(Q^{t_0(0)}(t_0^{2}(Q^{2}(Q^{2}(1,t_0^{[1]}(1)_*)))_*,t_0^{2}(Q^{2}(Q^{2}(1,t_0^{[1]}(1)_*))))_*,0),3)\)

\(\alpha(106.77) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q^{2}(1,t_0^{[1]}(1)_*)))_*,0),3)\)

\(\alpha(106.78) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q^{2}(1,t_0^{[1]}(1)_*)))_*,0),3)\)

\(\alpha(106.82) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(1,t_0^{[1]}(1)_*))_*,0),3)\)

\(\alpha(107.02) = Q(t_0^{2}(t_0^{2}(Q(Q(2,t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.03) = Q(t_0^{2}(Q^{t_0(0)}(t_0^{2}(Q^{2}(Q(2,t_0^{[1]}(1))))_*,t_0^{2}(Q^{2}(Q(2,t_0^{[1]}(1)))))_*,0),3)\)

\(\alpha(107.04) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q(2,t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.07) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q(1,Q(2,t_0^{[1]}(1)))))_*,0),3)\)

\(\alpha(107.08) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q^{2}(1,Q(2,t_0^{[1]}(1))_*)))_*,0),3)\)

\(\alpha(107.09) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(1,Q(2,t_0^{[1]}(1))_*))_*,0),3)\)

\(\alpha(107.11) = Q(t_0^{2}(Q^{2}(t_0^{2}(Q^{2}(2,t_0^{[1]}(1)_*))_*,t_0^{2}(Q^{2}(2,t_0^{[1]}(1)_*)))_*,0),3)\)

\(\alpha(107.12) = Q(t_0^{2}(Q^{t_0(0)}(t_0^{2}(Q(Q^{2}(2,t_0^{[1]}(1)_*)))_*,t_0^{2}(Q(Q^{2}(2,t_0^{[1]}(1)_*))))_*,0),3)\)

\(\alpha(107.13) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q^{2}(2,t_0^{[1]}(1)_*)))_*,0),3)\)

\(\alpha(107.14) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q^{2}(2,t_0^{[1]}(1)_*)))_*,0),3)\)

\(\alpha(107.16) = Q(t_0^{2}(t_0^{2}(Q(1,Q^{2}(2,t_0^{[1]}(1)_*)))_*,0),3)\)

\(\alpha(107.17) = Q(t_0^{2}(t_0^{2}(Q^{2}(1,Q^{2}(2,t_0^{[1]}(1)_*)_*))_*,0),3)\)

\(\alpha(107.18) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q^{2}(1,Q^{2}(2,t_0^{[1]}(1)_*)_*)))_*,0),3)\)

\(\alpha(107.19) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(1,Q^{2}(2,t_0^{[1]}(1)_*)_*))_*,0),3)\)

\(\alpha(107.21) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(2,t_0^{[1]}(1)_*))_*,0),3)\)

\(\alpha(107.31) = Q(t_0^{2}(t_0^{2}(Q^{Q(t_0(0),t_0(0))}(2,t_0^{[1]}(1)_*))_*,0),3)\)

\(\alpha(107.40) = Q(t_0^{2}(Q^{2}(t_0^{2}(Q(t_0^{[1]}(1),t_0^{[1]}(1)))_*,t_0^{2}(Q(t_0^{[1]}(1),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.41) = Q(t_0^{2}(t_0^{2}(Q^{2}(Q(t_0^{[1]}(1),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.42) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q(t_0^{[1]}(1),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.43) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q^{2}(1,Q(t_0^{[1]}(1),t_0^{[1]}(1))_*)))_*,0),3)\)

\(\alpha(107.44) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(1,Q(t_0^{[1]}(1),t_0^{[1]}(1))_*))_*,0),3)\)

\(\alpha(107.45) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q^{2}(2,Q(t_0^{[1]}(1),t_0^{[1]}(1))_*)))_*,0),3)\)

\(\alpha(107.46) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(2,Q(t_0^{[1]}(1),t_0^{[1]}(1))_*))_*,0),3)\)

\(\alpha(107.47) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q(Q(t_0^{[1]}(1)),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.48) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q^{2}(1,Q(Q(t_0^{[1]}(1)),t_0^{[1]}(1))_*)))_*,0),3)\)

\(\alpha(107.49) = Q(t_0^{2}(t_0^{2}(Q(1,Q^{2}(2,Q(Q(t_0^{[1]}(1)),t_0^{[1]}(1))_*)))_*,0),3)\)

\(\alpha(107.50) = Q(t_0^{2}(t_0^{2}(Q(t_0(0),Q(Q(t_0^{[1]}(1)),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.53) = Q(t_0^{2}(t_0^{2}(Q(Q^{2}(t_0^{[1]}(1)),t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(107.54) = Q(t_0^{2}(t_0^{2}(Q(1,Q(Q^{2}(t_0^{[1]}(1)),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.55) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(1,Q(Q^{2}(t_0^{[1]}(1)),t_0^{[1]}(1))_*))_*,0),3)\)

\(\alpha(107.56) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(2,Q(Q^{2}(t_0^{[1]}(1)),t_0^{[1]}(1))_*))_*,0),3)\)

\(\alpha(107.57) = Q(t_0^{2}(t_0^{2}(Q(t_0(0),Q(Q^{2}(t_0^{[1]}(1)),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.60) = Q(t_0^{2}(t_0^{2}(Q(Q^{t_0(0)}(t_0^{[1]}(1)),t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(107.79) = Q(t_0^{2}(t_0^{2}(Q(Q(1,t_0^{[1]}(1)),t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(107.80) = Q(t_0^{2}(t_0^{2}(Q(2,Q(Q(1,t_0^{[1]}(1)),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.81) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(1,Q(Q(Q(1,t_0^{[1]}(1))),t_0^{[1]}(1))_*))_*,0),3)\)

\(\alpha(107.82) = Q(t_0^{2}(t_0^{2}(Q(t_0(0),Q(Q(Q(1,t_0^{[1]}(1))),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.83) = Q(t_0^{2}(t_0^{2}(Q(t_0(0),Q(Q^{2}(Q(1,t_0^{[1]}(1))),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.84) = Q(t_0^{2}(t_0^{2}(Q(Q^{t_0(0)}(Q(1,t_0^{[1]}(1))),t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(107.89) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(Q(Q^{2}(1,t_0^{[1]}(1)_*),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.90) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(2,Q(Q^{2}(1,t_0^{[1]}(1)_*),t_0^{[1]}(1))_*))_*,0),3)\)

\(\alpha(107.91) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(2,Q(Q(Q^{2}(1,t_0^{[1]}(1)_*)),t_0^{[1]}(1))_*))_*,0),3)\)

\(\alpha(107.92) = Q(t_0^{2}(t_0^{2}(Q(Q^{2}(Q^{2}(1,t_0^{[1]}(1)_*)),t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(107.93) = Q(t_0^{2}(t_0^{2}(Q(t_0(0),Q(Q^{2}(Q^{2}(1,t_0^{[1]}(1)_*)),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(107.94) = Q(t_0^{2}(t_0^{2}(Q(Q^{t_0(0)}(Q^{2}(1,t_0^{[1]}(1)_*)),t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(107.99) = Q(t_0^{2}(t_0^{2}(Q(Q^{t_0(0)}(1,t_0^{[1]}(1)_*),t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(108.19) = Q(t_0^{2}(t_0^{2}(Q(1,Q(Q(Q(2,t_0^{[1]}(1))),t_0^{[1]}(1))))_*,0),3)\)

\(\alpha(108.20) = Q(t_0^{2}(t_0^{2}(Q^{t_0(0)}(2,Q(Q^{2}(Q(2,t_0^{[1]}(1))),t_0^{[1]}(1))_*))_*,0),3)\)

\(\alpha(108.21) = Q(t_0^{2}(t_0^{2}(Q(Q^{t_0(0)}(Q(2,t_0^{[1]}(1))),t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(108.23) = Q(t_0^{2}(t_0^{2}(Q(Q(1,Q(2,t_0^{[1]}(1))),t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(108.24) = Q(t_0^{2}(t_0^{2}(Q(Q^{t_0(0)}(Q(1,Q(2,t_0^{[1]}(1)))),t_0^{[1]}(1)))_*,0),3)\)

\(\alpha(108.25) = Q(t_0^{2}(t_0^{2}(Q(Q^{t_0(0)}(Q^{2}(1,Q(2,t_0^{[1]}(1))_*)),t_0^{[1]}(1)))_*,0),3)\)

Code changes (in this version)
The javascript code is not complete. I plan to add some improvements and fix a number of errors.

Improvement #1: The output code should use base 1 instead of 2. I think this is more interesting and a little easier to make comparisons between the Quantum output function and a similar FGH function. Refer to my Alpha Function blog for examples. This is easy to change but using base 2 makes the next version of the code easier to debug as it is being developed.

Improvement #2: The algorithm will be calibrated to approximately:

\(\alpha(2^{512}) = Q(1,0,1) = Q(t_{Q(1,1)}(0),1) = Q(t_3(0),1)\)

where

\(t_3(0) = t_2(1,0_{[t_2(0)]})\)

The calibration algorithm will need be adjusted when the full code has been completed.

Error #1: The code is not complete. The algorithm does not generate t function values of \(t_1(0)\) or greater. This will be the focus of the next version.

Indexing & Summation Notation
This version introduces Indexing Notation. I use notation that is not in common use, and I have added 'Indexing' and 'Summation'notation. Here is a complete explanation of the notation used in the Quantum function examples in this blog and in the spreadsheet.

\(Q(n) = n + 1\)

Decremented Function \(C\) is short-hand notation. For any arbitrary function:

\(Q^{c + 1}(n)\) then \(C = Q^{c}(n)\)

\(Q^{c + 1}(n) = Q(C) = Q(Q^c(n))\)

\(Q(1,n) = Q^{Q(n)}(n)\)

Recursion Parameter Subscript \(*\) identifies how recursion applies to multi-parameter function.

\(Q^{c + 1}(1,n_*) = Q(1,C) = Q(1,Q^c(1,n_*))\)

\(Q^{c + 1}(1_*,n) = Q(C,n) = Q(Q^c(1,n_*),n)\)

\(Q(c + 1,n) = Q^C(C,n_*)\)

\(Q(t_0(0),n) = Q(n,n)\)

\(t_0(c + 1) = Q^C(C_*,C)\)

Indexing Notation is shorthand notation (index) for a function within a nested stack.

\(t_0^5(0) = t_0(t_0(t_0(t_0(t_0(0)))))\)

then indexing notation is useful shorthand for these nested functions.

\(t_0^5(Q(t_0^{[4]}(0))) = t_0(Q(t_0(t_0(t_0(t_0(0))))))\)

\(t_0^5(Q^2(t_0^{[2]}(0))) = t_0(t_0(t_0(Q^2(t_0(t_0(0))))))\)

\(t_0^5(Q(t_0^{[4]}(Q^2(t_0^{[2]}(0))))) = t_0(Q(t_0(t_0(Q^2(t_0(t_0(0)))))))\)

and indexing notation is used by the \(t\) function:

\(t_0^{c + 1}(x) = t_0(C) = D\)

then

\(Q^D(D_*,D) = t_0^{c + 1}(Q(t_0^{[c]}(x)))\)

Parameter Subscript Brackets is shorthand for functions with multiple parameters:

\(t_0(x,0_{[2]}) = t_0(x,0,0)\)

\(t_0(x,y_{[2]}) = t_0(x,y_1,y_2)\)

\(t_0(x,0_{[2]},y_{[3]},z) = t_0(x,0,0,y_1,y_2,y_3,z)\)

Leading Zeros Assumption applies and all leading zeroes can be ignored:

\(t_0(0_{[x]},0_{[2]},y_{[3]},1) = t_0(0_{[x + 2]},y_{[3]},1) = t_0(y_{[3]},1) = t_0(y_1,y_2,y_3,1)\)

this notation is used by the \(t\) function:

\(t_0(1,0_{[c + 1]}) = t_0^C(1_*,0_{[c + 1]})\)

e.g.

\(t_0(1,0) = t_0^C(1_*,0_{[0]}) = t_0^C(1) = t_0^{t_0(1)}(1)\)

Summation Notation introduces summation functions \(t_0(s)\) for use with Indexing Notation:

\(t_0(s)(t_0(1,0)) = t_0(1)\)

\(t_0(s)(t_0(1,c + 1)) = t_0(1,c) + C = t_0(1,c) + t_0(s)(t_0(1,c))\)

this notation is used by the \(t\) function:

\(t_0(1,c) = t_0^{t_0(s)(t_0(1,c))}(1) = D\)

then

\(Q^D(D_*,D) = t_0^{t_0(s)(t_0(1,c))}(Q(t_0^{[t_0(s)(t_0(1,c)) - 1]}(1)))\)

\(t_0(1,c + 1) = t_0^C(C) = t_0^{t_0(1,c)}(t_0(1,c))\)

\(= t_0^{t_0(1,c)}(t_0^{t_0(s)(t_0(1,c))}(1)) = t_0^{t_0(1,c) + t_0(s)(t_0(1,c))}(1) = t_0^{t_0^{t_0(s)(t_0(1,c + 1))}(1)\)

\(t_0^{c + 1}(1,0_*) = t_0(1,C) = D\)

then

\(Q^D(D_*,D) = t_0^{t_0(s)(D)}(Q(t_0^{[t_0(s)(D) - 1]}(1)))\)

Further References
Further references to relevant blogs can be found here: User:B1mb0w