I looked at it from the "other side": get all the variations with repetition of the set of triangular numbers less than 80. Then compute their sums. Store these sums in a an array (yes it will overwrite previous results, but we only need one solution for each number) and print the results for the numbers upto 80.

use Modern::Perl;
use Algorithm::Combinatorics qw /variations_with_repetition/;
use List::Util qw /sum/;

my @triangulars;
push @triangulars, $_ * ( $_ + 1 ) / 2 for 0 .. 12;

my $iterator = variations_with_repetition( \@triangulars, 3 );

my @results;
$results[ sum @$_ ] = join ' + ', @$_ while $_ = $iterator->next;

say "$_ = $results[$_]" for 0 .. 80;
[download]

Output:

0 = 0 + 0 + 0
1 = 1 + 0 + 0
2 = 1 + 1 + 0
3 = 3 + 0 + 0
4 = 3 + 1 + 0
5 = 3 + 1 + 1
6 = 6 + 0 + 0
7 = 6 + 1 + 0
8 = 6 + 1 + 1
9 = 6 + 3 + 0
10 = 10 + 0 + 0
11 = 10 + 1 + 0
12 = 10 + 1 + 1
13 = 10 + 3 + 0
14 = 10 + 3 + 1
15 = 15 + 0 + 0
16 = 15 + 1 + 0
17 = 15 + 1 + 1
18 = 15 + 3 + 0
19 = 15 + 3 + 1
20 = 10 + 10 + 0
21 = 21 + 0 + 0
22 = 21 + 1 + 0
23 = 21 + 1 + 1
24 = 21 + 3 + 0
25 = 21 + 3 + 1
26 = 15 + 10 + 1
27 = 21 + 6 + 0
28 = 28 + 0 + 0
29 = 28 + 1 + 0
30 = 28 + 1 + 1
31 = 28 + 3 + 0
32 = 28 + 3 + 1
33 = 21 + 6 + 6
34 = 28 + 6 + 0
35 = 28 + 6 + 1
36 = 36 + 0 + 0
37 = 36 + 1 + 0
38 = 36 + 1 + 1
39 = 36 + 3 + 0
40 = 36 + 3 + 1
41 = 28 + 10 + 3
42 = 36 + 6 + 0
43 = 36 + 6 + 1
44 = 28 + 15 + 1
45 = 45 + 0 + 0
46 = 45 + 1 + 0
47 = 45 + 1 + 1
48 = 45 + 3 + 0
49 = 45 + 3 + 1
50 = 28 + 21 + 1
51 = 45 + 6 + 0
52 = 45 + 6 + 1
53 = 28 + 15 + 10
54 = 45 + 6 + 3
55 = 55 + 0 + 0
56 = 55 + 1 + 0
57 = 55 + 1 + 1
58 = 55 + 3 + 0
59 = 55 + 3 + 1
60 = 45 + 15 + 0
61 = 55 + 6 + 0
62 = 55 + 6 + 1
63 = 45 + 15 + 3
64 = 55 + 6 + 3
65 = 55 + 10 + 0
66 = 66 + 0 + 0
67 = 66 + 1 + 0
68 = 66 + 1 + 1
69 = 66 + 3 + 0
70 = 66 + 3 + 1
71 = 55 + 15 + 1
72 = 66 + 6 + 0
73 = 66 + 6 + 1
74 = 45 + 28 + 1
75 = 66 + 6 + 3
76 = 66 + 10 + 0
77 = 66 + 10 + 1
78 = 78 + 0 + 0
79 = 78 + 1 + 0
80 = 78 + 1 + 1
[download]

---

---

• Are you posting in the right place? Check out Where do I post X? to know for sure.
• Posts may use any of the Perl Monks Approved HTML tags. Currently these include the following:

  <code> <a> <b> <big> <blockquote> <br /> <dd> <dl> <dt> <em> <font> <h1> <h2> <h3> <h4> <h5> <h6> <hr /> <i> <li> <nbsp> <ol> <p> <small> <strike> <strong> <sub> <sup> <table> <td> <th> <tr> <tt> <u> <ul>

• Snippets of code should be wrapped in <code> tags not <pre> tags. In fact, <pre> tags should generally be avoided. If they must be used, extreme care should be taken to ensure that their contents do not have long lines (<70 chars), in order to prevent horizontal scrolling (and possible janitor intervention).
• Want more info? How to link or How to display code and escape characters are good places to start.