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 further golfing forthcoming.
$A=1111;1..($n=4)|%{$A;[string]$B=[math]::pow($A,2);$A=$B.substring(0,4)}
$n
is n in xn-1
Explanation and further golfing forthcoming.