EDIT: For some reason, a few numbers such as 1035 are causing errors (Can't take sqrt of -1, <STDIN> line 1.) Going to have to hunt down the bug and fix it...

EDIT: Just needed a slight change to the first line after BING to prevent tget calls for 0:

use strict;
use warnings;

my ($i, $j, $inum, $jnum, $left);

$| = 1; print "Enter the number to check : ";
my $inp = <STDIN>;

BING:
for ($i = tget($inp-2); $i; $i--) {
    $inum = tmake($i);
    $left = tget($inp - $inum - 1) + 1;
    for ($j = 1; $j < $left; $j++) {
        $jnum = tmake($j);
        last BING if (tis($inp - $inum - $jnum));
    }
}
print "       Your triangles are : $inum $jnum "
      . ($inp - $inum - $jnum) . "\n";

sub tget { return int(sqrt(1 + $_[0] * 2) - .5); }
sub tmake { return (.5 * $_[0] * ($_[0] + 1)); }
sub tis { my $n = shift; return ($n == tmake(tget($n))); }
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Incidently, I ran a looped version of this for all numbers 3-100000, and only the following could not be made with sums of three triangles:

4, 6, 11, 20, 29

As far as I can tell, all integers after 29 can be made with at least one sum of three triangles. The number of sums increases as you go along.