User blog comment:Wythagoras/SuperJedi's X function/@comment-4224897-20141113192733/@comment-10429372-20150430184833

(Spamming out comments here because I haven't any scratch paper near)

I think that it even can be done in O(N):  A sketch:

A(a,b,c,d,e) = A(a+10^d, b, c+1, d-1, e)

Which amounts to 20 symbols, *2 (special case d=0 and we switch form tapes) *c (number of colors)

So I think that, provided that c≤10, the following bound should hold:

BB(s,c) < X(42sc) [lol, 42]