I'm not sure quite how you might take advantage of it for optimisation, but the possible decompositions are restricted by the mod 3 equivalences. That is: T_n == 1 (mod 3) when n == 1 (mod 3)
T_n == 0 (mod 3) otherwise
so if we split the T_n sequence into A_n (== 0) and B_n (== 1) the possible decompositions are restricted such that: if n == 0 (mod 3), require A A A or B B B
if n == 1 (mod 3), require A A B
if n == 2 (mod 3), require A B B
You can get similar restrictions by considering other prime moduli, but 3 is likely to be the most beneficial because it has a shorter than possible cycle in T_n.
Also, a word of warning on your p_tri() routine  the final result relies on comparing a floating point number for equality, normally considered a bad idea. It would be preferable to do the test instead by calculating from $t back up to the last known integer value, something like: sub p_tri {
my $num = shift;
my $x = 8 * $num + 1;
my $t = int((sqrt($x) + 1)/2);
return +((2 * $t  1) * (2 * $t  1) == $x) ? 0 : $t;
}
Hugo
Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
Read Where should I post X? if you're not absolutely sure you're posting in the right place.
Please read these before you post! —
Posts may use any of the Perl Monks Approved HTML tags:
 a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
Outside of code tags, you may need to use entities for some characters:

For: 

Use: 
 &   & 
 <   < 
 >   > 
 [   [ 
 ]   ] 
Link using PerlMonks shortcuts! What shortcuts can I use for linking?
See Writeup Formatting Tips and other pages linked from there for more info.

