And further: $$\hardy(a_2, a_1, a_0,n) = \begin{cases} \begin{cases} \begin{cases}
n, &a_2=0 \\ \hardy(a_2-1, n, n, n), & a_2>0 \end{cases}, &a_1=0 \\ \hardy(a_2, a_1-1, n, n), & a_1>0 \end{cases}, & a_0=0 \\ \hardy (a_2, a_1, a_0-1, n+1), &a_0>0 \end{cases}$$