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Powershell, 73 bytes

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

$n is n in x_n-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
}