What I've settle on is shown below. It won't remove a single blank line at the beginning of the file, but I don't need that. It may not be the most efficient way to do this, but it make sense to me. I learned a bunch. Thanks!

#!/usr/bin/perl
while (<>) {
  $last = $this;
  $this = $next;
  $next = $_;
  $lastblank = ($last =~ (/^\s*$/));
  $thisblank = ($this =~ (/^\s*$/));
  $nextblank = ($next =~ (/^\s*$/));

  print $this unless (! $lastblank) && ($thisblank) && (! $nextblank);
}
print $next unless ($nextblank) && (! $thisblank);
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