L $=\{$a

ka

k

$|$ k in N$0\}$ is not regular

We choose the word x $=$ a

na

n

$.$ Then $|$x$|=2$n $>=$ n and x in L$.$ We consider all decompositions

x $=$ uvw where $|$v$|>=1,|$uv$|<=$ n:

$1)$ u $=$ a

k

$,$ v $=$ a

$`$

$,$ w $=$ a

n$$k$`$a

n where k $>=0,`>=1,$ k $+`<=$ n