Just off the top of my head:
In the inner while loop, it looks like you're testing each $i, $j to see if it's triangular (although I must confess, I don't really understand p_tri). Since we can enumerate triangle numbers (if Jobby is to be believed), why not iterate through those?
Edit: Are you sure you're getting correct solutions? It looks like $prev starts out as a triangular number, but then you just decrement it. (Aside: if this algorithm gives you consistently correct solutions, then the greatest triangular number lower than n is always present in n's triangular decomposition. I think.) Man, I'm an idiot. Since when is $prev a trinum?
Edit 2: Oh, I get it, p_tri returns the rank of the previous triangular number, not the number itself... although it returns zero when given a triangular number, which is weird but useful.
Edit 3: Here's the start of an implementation. It's not as fast as the code posted above (by about a factor of two, if Unix time is to be trusted), probably because it calculates a lot of trinums and makes a lot of function calls. I'm posting it more or less as a proof of concept.
Edit 4: Inlining the calls to &trinum results in slightly faster code than Limbic~Region's. Code updated, benchmarks added.
#! /usr/bin/perl w
use strict;
# trinum(n) returns the nth triangular number
sub trinum { my ($n) = @_; return $n * ($n+1) * 0.5; }
# prev_trinum(n) returns the RANK OF the greatest triangular number le
+ss
# than n.
# Code blatantly ripped off from Limbic~Region [id://399054]
sub prev_trinum
{
my $num = shift;
my $x = ( sqrt( 8 * $num + 1 ) + 1 )/ 2;
my $t = int $x;
return $t == $x ? 0 : $t;
}
# trinum_decomp(n) tries to find a threetriangularnumber decompositi
+on
# of n. Based on L~R's method from the post cited above, but
# enumerates trinums rather than guessing.
sub trinum_decomp
{
my ($n) = @_;
my $prev = &prev_trinum($n);
return ($n, 0, 0) unless $prev;
while($prev) {
my $triprev = ($prev * $prev + $prev)/2;
my $diff = $n  $triprev;
my @tail = &twonum_decomp($diff);
if(defined $tail[0]) {
return ($triprev, @tail);
}
$prev;
}
warn "Can't find trnum decomp for $n\n";
return (1, 1, 1); # ugly
}
# twonum_decomp(n) tries to find a twotriangularnumber decomposition
# of n. If such a decomposition does not exist, returns undef.
sub twonum_decomp
{
my ($n) = @_;
my $prev = &prev_trinum($n);
return ($n, 0) unless $prev;
while($prev) {
my $triprev = ($prev * $prev + $prev)/2;
my $i = 1; my $tri_i = ($i * $i + $i)/2;
do {
if($tri_i + $triprev == $n) {
return ($tri_i, $triprev);
}
$i++; $tri_i = ($i * $i + $i)/2;
} while($triprev + $tri_i <= $n);
$prev;
}
return undef;
}
my $target = $ARGV[0]  314159;
print join(',', &trinum_decomp($target));
__END__
mjolson@riga:~/devel/scratch
Wed Oct 1318:38:42 583 >time ./trinum 987654321
987567903,14028,72390
real 0m0.089s
user 0m0.060s
sys 0m0.000s
mjolson@riga:~/devel/scratch
Wed Oct 1318:18:25 578 >time ./limbic_trinum 987654321
987567903, 14028, 72390
real 0m0.106s
user 0m0.090s
sys 0m0.000s

Yours in pedantry,
F
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