Powershell, 73 bytes
$A=1111;1..($n=4)|%{$A;[string]$B=[math]::pow($A,2);$A=$B.substring(0,4)}
$n
is n in xn-1
Explanation and exploded code:
$A=1111 #starting number
$n=4 #n in formula
for($i=0; $i -lt $n;$i++) #loop n times
{
$A #print $A
[string]$B=[math]::pow($A,2) #create a new string $B and set it to $A raised to the power of 2
$A=$B.substring(0,4) #set $A to the first 4 characters of $B
}