User blog comment:B1mb0w/Fundamental Sequences/@comment-26428969-20151104162536

the reason why "phi(a+1,b+1)[0] = phi(a+1,b)+1" is that phi(a+1,X) is defined as the Xth fixed point of "Y -> phi(a,Y)", so phi(a+1,b+1) is the limit of phi(a+1,b), phi(a,phi(a+1,b)+1) (you need the +1 because phi(a,phi(a+1,b)) = phi(a+1,b) by definition!), phi(a,phi(a,phi(a+1,b)+1)), ... and the +1 is needed only at the beginning so c=0 in "φ(α,φ(α+1,0)[n]+c)"