User blog comment:Mh314159/YIP notation/@comment-35470197-20190704230823

We have \begin{eqnarray*} f_a(b) & \sim & F_{2a}(b) \\ [a] & \sim & F_{2a}(a) = F_{2 \omega}(a) \\ [a,1] & \sim & F_{2 \omega}^{F_{2 \omega}(a)}(a) \sim F_{2 \omega + 1}(F_{2 \omega}(a)) \\ X_0(a,2) & = & [a,1] \sim F_{2 \omega + 1}(F_{2 \omega}(a)) \\ X_1(a,2) & = & F_{2 \omega}^{X_0(a,1)}(a) \sim F_{2 \omega + 1}^2(F_{2 \omega}(a)) \\ X_k(a,2) & \sim & F_{2 \omega + 1}^{k+1}(F_{2 \omega}(a)) \\ [a,2] & = & X_{[a,1]}(a,2) \sim F_{2 \omega + 2}(F_{2 \omega + 1}(F_{2 \omega}(a))) \\ X_0(a,3) & = & [a,2] \sim F_{2 \omega + 2}(F_{2 \omega + 1}(F_{2 \omega}(a))) \\ X_n(a,3) & = & (F_{2 \omega + 1} \circ F_{2 \omega})^n(X_0(a,3)) \sim (F_{2 \omega + 1} \circ F_{2 \omega})^n(F_{2 \omega + 2}(F_{2 \omega + 1}(F_{2 \omega}(a)))) \ [a,3] & = & X_{[a,2]}(a,3) \sim (F_{2 \omega + 1} \circ F_{2 \omega})^{F_{2 \omega + 2}(F_{2 \omega + 1}(F_{2 \omega}(a)))}(F_{2 \omega + 2}(F_{2 \omega + 1}(F_{2 \omega}(a)))) \\ & \sim & F_{2 \omega + 2}^2(a), [a,n] & \sim & F_{2 \omega + 2}^{n-1}(a) \\ [n,n] & \sim & F_{2 \omega + 3}(n), \end{eqnarray*} where \(F\) denotes the FGH. Theredore the limit is \(2 \omega + 3\) in FGH.