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]

CountZero

A program should be light and agile, its subroutines connected like a string of pearls. The spirit and intent of the program should be retained throughout. There should be neither too little or too much, neither needless loops nor useless variables, neither lack of structure nor overwhelming rigidity." - The Tao of Programming, 4.1 - Geoffrey James

In reply to Re: Write any natural number as sum of three triangular numbers by CountZero
in thread Write any natural number as sum of three triangular numbers by ambrus

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