You're almost there:

#!/usr/bin/perl

use strict;
use warnings;

my $count;
do {$_ = <DATA>; print} until /formulae/ && ++ $count == 2;

__DATA__
One line
two line
formulae
blue line
red line
formulae
more line
more line
formulae
final line
[download]

That will print:

One line
two line
formulae
blue line
red line
formulae
[download]

Abigail