Powershell, 73 bytes
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    $A=1111;1..($n=4)|%{$A;[string]$B=[math]::pow($A,2);$A=$B.substring(0,4)}

 `$n` is **n** in **x<sub>n-1</sub>**

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
    }