User blog comment:Ikosarakt1/Unusual number naming scheme/@comment-25418284-20140429200600/@comment-5150073-20140430125020

''Although it's significantly dirtier, you could just enumerate all the possible sequences by length and lexicographical order. So zeroarrowal = 3v3, onearrowal = 3^3, twoarrowal = 3vv3, threearrowal = 3v^3 ...''

That defines exactly the same sequence. Twoarrowal = $$3 \downarrow\downarrow 3 = 3 \uparrow\downarrow 3$$, there is no matter whether the first arrow up or down.

''* googol = 10^100 What's the logic for this naming anyways?
 * kookol = 10vv100
 * ɟooɟol = 10v^100
 * coocol = 10^v100
 * ɖooɖol = 10^^100
 * ʈooʈol = 10vvv100
 * doodol = 10vv^100
 * tootol = 10v^v100
 * d̪ood̪ol = 10v^^100
 * t̪oot̪ol = 10^vv100
 * ɣooɣol = 10X10, where X is encoded by 100''