Another one, with rotating:

#!/usr/bin/perl

# https://perlmonks.org/?node_id=1227147

use warnings;
use strict;

my $short_pattern = 'FTTF';

my @result = grep { 
                $short_pattern = ( chop $short_pattern ) . $short_patt
+ern;
                $short_pattern ge 'T' 
                } 0 .. 12;

print "@result\n";
[download]

P.S. note a wrong rotation direction. So, the pattern should be reversed before, if it's not a palindrome.