As a one-liner

perl -ne "$n = $.if /line 05/; print if $n and $. == ( $n + 4 )" numbe
+red.dat
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As a program

But if the pattern can appear more than once, it gets tricky, since it may be that n = 5 (say) but the pattern is seen a second time only 3 lines after the first. Then you sort of need to remember which line numbers will need to be printed out, and print them as you come to them.

Here's my solution (tested):

  my $pattern = qr/match this string!/;
  my $n = 4;

  my @li;
  while(<>)
  {
    if ( @li && $. == $li[0] )
    {
      print;
      shift @li;
    }
    /$pattern/ and push @li, $. + $n;
  }
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while (<>) {
 next unless /$pattern/;
 for (1..$n){
    undef = (<>);
 }
 print;
 last;
}
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#generator
perl -le " print qq,line $_, for 1..100 "

#the proggie (prints the 6th line after /line 1/ is detected
perl -ne " /line 1/ ..  $a++ == 6 && print "

# sample run
perl -le " print qq,line $_, for 1..100 " | perl -ne " /line 1/ ..  $a
+++ == 6 && print "

#yields
line 7
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____________________________________________________
** The Third rule of perl club is a statement of fact: pod is sexy.

open (FILE '<filename');
$_ = join('',<FILE>);
close (<FILE>);
my ($result) = m!$match(?:.*?$/){$Nth}(.*?$/)!s;
print $result;
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open (FILE '<filename');
$_ = join('',<FILE>);
while ( m!$match(?:.*?$/){$Nth}(.*?$/)!sg) {
  print $1;
#  pos = length ($match) + length ($`);
}
[download]

Dingus