User blog comment:WaxPlanck/k function/@comment-25601061-20171224185209/@comment-28606698-20171226195252

Seems WaxPlanck used the following definition of his k-function

$$k(n)=n \uparrow^n n=H_{n+2}(n,n)$$

where

$$ H_n(a,b) = \begin{cases} b + 1 & \text{if } n = 0 \\ a-1 & \text{if } n = 1 \text{ and } b = -1 \\ 0 & \text{if } n = 2 \text{ and } b = 0 \\ 1 & \text{if } n \ge 3 \text{ and } b = 0 \\ H_{n-1}(a, H_n(a, b - 1)) & \text{otherwise} \end{cases} $$

Then

$$k(-2)=H_{0}(-2,-2)=-2+1=-1$$

$$k(-1)=H_{-1+2}(-1,-1)=H_{1}(-1,-1)=-1-1=-2$$