Triangular numbers
A triangular number is the running total of the whole numbers up to some point — 1, 3, 6, 10, 15 and so on — so called because that many dots arrange exactly into a triangle.
Triangular numbers count the dots in a growing triangle: add 1, then 2, then 3, and each running total is the next triangular number. They are a purely mathematical grouping. A few members also carry unrelated cultural or everyday significance (10 and the decimal base, 21 and a coming-of-age threshold, 666 and its religious associations), but that significance has nothing to do with being triangular and is covered on each number's own page.
Triangular numbers as triangles
A triangular number is what you get by adding up the whole numbers in order and stopping somewhere. Add 1 and you have 1. Add 2 more and you have 3. Add 3 more and you have 6. Each running total is the next triangular number, and each one is exactly the number of dots needed to fill a triangle one row deeper than the last.
The formula, and why it works
The nth triangular number is n × (n + 1) ÷ 2. The 17th is 17 × 18 ÷ 2 = 153; the 10th is 10 × 11 ÷ 2 = 55.
The reason is worth seeing rather than memorising. Write the numbers 1 to 10 in a row, then write them again underneath in reverse. Every column now adds to 11, and there are 10 columns — 110 in total. But that counted every number twice, so the real total is 110 ÷ 2 = 55. Nothing about this depends on 10, which is why the formula holds for every n.
Where triangular and square numbers meet
A number can be both triangular and square, but it is unusual. In this catalogue only 1 and 36 qualify — 36 is 6 × 6, and it is also 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8. It is the smallest number above 1 that is both.
These are called square triangular numbers, and they become scarce very fast: after 36 the next is 1225, then 41616. There is a neat companion fact in the other direction — add any two consecutive triangular numbers and you always get a square. 6 + 10 = 16, and 10 + 15 = 25.
Triangular numbers in the catalogue
1
3
6
10
15
21
28
36
45
55
66
78
91
105
120
153
171
190
210
231
253
276
300
325
351
378
406
465
496
528
561
595
630
666
741
903
990
1035
1081
1176
1225
1275
1378
1431
1485
1596
1653
1711
1770
1830
1891
1953
2016
3003
5778
8,128
17,020
40,755
42,195
Common questions
What is a triangular number?
It is the running total of the whole numbers counted up to some point. 1 is the first, 1 + 2 = 3 is the second, 1 + 2 + 3 = 6 is the third, and so on. The name comes from the fact that exactly that many dots can be arranged into a filled triangle.
How do I check whether a number is triangular?
Multiply it by 8 and add 1. If the result is a perfect square, the number is triangular. For 15: 8 × 15 + 1 = 121, which is 11², so 15 is triangular. For 16: 8 × 16 + 1 = 129, which is not a perfect square, so 16 is not.
Is there a formula for the nth triangular number?
Yes — n × (n + 1) ÷ 2. The 17th triangular number is 17 × 18 ÷ 2 = 153. The formula works because pairing the first and last terms, second and second-last, and so on, always gives the same total.
Can a number be both triangular and square?
Yes, though it is rare. In this catalogue only 1 and 36 are both. 36 is the smallest number above 1 with that property — it is 6 × 6 and also 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8. Such numbers are called square triangular numbers, and they thin out very quickly (the next is 1225).
Does being triangular mean a number is significant in other ways?
No. It is a purely mathematical property. Some members of this hub do carry separate significance — 10 underpins the decimal system, 21 marks a coming-of-age threshold in several countries — but that is unrelated to their triangular status and is covered on their own profile pages.