Beyond the steady climbers — the arithmetic sequences that add a fixed step, and the geometric ones that multiply by a fixed factor — live a handful of famous special sequences with patterns so pretty they've been admired for thousands of years. The square numbers, the triangular numbers, the cube numbers, and the celebrated Fibonacci sequence each hide a neat rule — and each has a surprise waiting behind it.

These sequences don't follow a single term-to-term rule like "add 3." Instead each one is built from a picture or a small trick. Once you spot the pattern you can keep going forever — and you'll start noticing these numbers hiding in sunflowers, handshakes, and stacks of dots.

Triangular numbers

Stack dots into a triangle: one dot, then a row of two beneath it, then a row of three… Counting all the dots gives the triangular numbers:

Each one adds the next whole row, so the gaps grow by one each time: +2, then +3, then +4. This is one example of an nth-term rule that isn't linear — the step keeps changing. There is a lovely shortcut, though: the nth triangular number is

(That's the arithmetic-series sum 1 + 2 + \dots + n in disguise — exactly the same pairing trick Gauss used.)

Watch the triangle grow one row at a time. Step through it.

Worked example — the 10th triangular number

You could add 1 + 2 + 3 + \dots + 10 by hand, but the formula is instant. With n = 10:

So the 10th triangular number is 55 — and, as we'll see, that's also the number of handshakes in a room of eleven people. Notice the little check: one of n and n+1 is always even, so n(n+1)/2 is always a whole number.

Square numbers

Now arrange the dots in a square instead — a 1\times1 square, then 2\times2, then 3\times3. Counting the dots gives the square numbers:

Each one is a whole number multiplied by itself — using index notation, the nth square number is n^2. So the 5th square number is 5^2 = 25.

Here's a hidden gem: the gaps between square numbers are 3, 5, 7, 9, \dots — the odd numbers, in order. Adding up the first few odd numbers always lands you on a perfect square (1 + 3 = 4, 1 + 3 + 5 = 9).

Cube numbers

Stack the dots into a solid block instead — 1\times1\times1, then 2\times2\times2, then 3\times3\times3 — and you count the cube numbers:

The nth cube number is n^3 = n\times n\times n. So the 4th cube number is 4^3 = 64. Their differences are 7, 19, 37, \dots — no constant gap in sight, which is the giveaway that a sequence is something more interesting than plain arithmetic.

The Fibonacci sequence

The Fibonacci sequence needs no picture — just a rule: start with two 1s, then each new term is the sum of the previous two.

Check it: 1+1=2, 1+2=3, 2+3=5, 3+5=8. To continue, just add the last two numbers you have. It's a term-to-term rule that reaches back two steps instead of one — which is why the differences never settle into a pattern.

Worked example — name that sequence

You're handed 2,\ 6,\ 12,\ 20,\ 30,\ \dots Is it arithmetic, geometric, or something special? Test the differences first:

The differences 4, 6, 8, 10 are not constant, so it isn't arithmetic. The ratios 6/2 = 3 and 12/6 = 2 aren't constant either, so it isn't geometric. But look at the differences again — they climb by 2 each time (a constant second difference), a sure sign of a special sequence. In fact 2, 6, 12, 20, \dots = n(n+1): twice the triangular numbers. The next term is 30 + 12 = 42.

Worked example — handshakes at a party

Six friends meet and everyone shakes everyone else's hand exactly once. How many handshakes?

Line them up: the 1st person shakes the other 5, the 2nd shakes the 4 remaining new hands, the 3rd shakes 3, and so on — 5 + 4 + 3 + 2 + 1. That's a triangular number! Using T_n = \tfrac{n(n+1)}{2} with n = 5:

In general, n people make \tfrac{n(n-1)}{2} handshakes — the triangular numbers, showing up at every party.

See it explained

Khan Academy builds triangular numbers from dots, just like the diagram above.