User blog comment:B1mb0w/Strong D Function/@comment-11227630-20150627053528

Your function is not so strong.

I get \(D(1,n,n)\approx f_{\omega+1}(n^2)\) then \(D(2,0,n)\approx f_{\omega+2}(n+2)\), \(D(2,1,n)\approx f_{\omega+2}(n+4)\), and \(D(2,n,n)\approx f_{\omega+2}(n^2)\).

Then \(D(3,0,n)\approx f_{\omega+3}(n)\), \(D(3,n,n)\approx f_{\omega+3}(n^2)\), \(D(4,0,n)\approx f_{\omega+4}(n)\) and so on.

So \(D(n,n,n)\approx f_{\omega2}(n)\) and \(D(1,0,0,n)\approx f_{\omega2+1}(n)\). \(D(1,0,n,n)\approx f_{\omega2+1}(n^2)\) and \(D(1,n,n,n)\approx f_{\omega2+1}(n^3)\).

Then \(D(2,0,0,n)\approx f_{\omega2+2}(n)\) and \(D(2,n,n,n)\approx f_{\omega2+2}(n^3)\); \(D(3,0,0,n)\approx f_{\omega2+3}(n)\) and \(D(3,n,n,n)\approx f_{\omega2+3}(n^3)\).

So \(D(n,n,n,n)\approx f_{\omega3}(n)\) and \(D(1,0,0,0,n)\approx f_{\omega3+1}(n)\). And further, \(D(n,n,n,n,n)\approx f_{\omega4}(n)\) and \(D(n,n,n,n,n,n)\approx f_{\omega5}(n)\), etc.

Finally \(D(\underbrace{n,n,\cdots n,n}_{n})\approx f_{\omega^2}(n)\), which is comparable to Conway's chained arrow notation.