Figurate Numbers Calculator

Differences define the family

Every figurate sequence is a running total of something, and that something is its difference pattern.

Triangular numbers are the running totals of 1, 2, 3, 4, … so their differences are the counting numbers. The nth is n(n + 1)/2, which Gauss famously worked out as a schoolchild by pairing the ends.

Square numbers have the odd numbers as differences: 1, 4, 9, 16 differ by 3, 5, 7. So 1 + 3 + 5 + … + (2n − 1) = n², which the Greeks proved by drawing L-shaped borders around a square — each border is the next odd number.

Tetrahedral numbers are running totals of the triangular numbers: a stack of triangular layers, which is how cannonballs and oranges are piled.

Seeing the differences is more useful than seeing the picture, because the differences generalise and the picture does not — there is no drawing for a four-dimensional figurate number, and the difference rule works anyway.

Is a number triangular?

There is a closed test, and it needs no searching: N is triangular exactly when 8N + 1 is a perfect square.

It comes straight from the formula. N = n(n+1)/2 rearranges to n² + n − 2N = 0, and the quadratic formula gives n = (−1 + √(8N + 1))/2. That is a whole number exactly when the square root is.

So 28 is triangular because 8×28 + 1 = 225 = 15², and the index is (15 − 1)/2 = 7. 30 is not, because 241 is not a square.

This page uses an integer square root on arbitrary-precision integers, so the test is exact for numbers far beyond what a double could hold — and the test suite checks it against a direct search over every value to 2,000, because a closed test that disagreed with a search would be worse than no test at all.

How the families overlap

The families are not disjoint, and the overlaps are more interesting than the definitions.

Every hexagonal number is triangular. The nth hexagonal number is the (2n − 1)th triangular one, so the hexagonals are exactly the odd-indexed triangulars. The reverse is not true.

Numbers that are both triangular and square are rare: 1, 36, 1225, 41616. They come from a Pell equation, and there are infinitely many, each roughly 34 times the last.

Both tetrahedral and square is rarer still: only 1, 4 and 19600. Proving there are no others took until 1878.

Every even perfect number is triangular. 6, 28, 496 and 8128 are all in the sequence, and it is not a coincidence — it follows from their form 2^(p−1)(2^p − 1), which is the triangular number of index 2^p − 1.

Where they turn up

Triangular numbers appear far more often than their definition suggests.

Handshakes. The number of pairs among n people is the (n−1)th triangular number. So is the number of edges in a complete graph, and the number of comparisons a naive duplicate check makes.

Pool balls and bowling pins. 15 and 10 are triangular, which is why those arrangements are neat.

Pascal’s triangle. The third diagonal is the triangular numbers and the fourth is the tetrahedral ones — they are binomial coefficients wearing a different name.

Pentagonal numbers appear in Euler’s pentagonal number theorem, which gives a startlingly efficient recurrence for counting partitions. Nobody would guess that from the geometry.

Where it gets used

Recognising a sequence. Given 1, 3, 6, 10, the difference pattern identifies the family immediately, which is the first step in most sequence puzzles.

Counting arrangements. Objects stacked in triangles or pyramids, and pairwise counts in combinatorics.

Algorithm analysis. A nested loop where the inner one shortens each time runs a triangular number of iterations, which is where the n²/2 in a bubble sort comes from.

Number theory. As a route into Pell equations, partitions and perfect numbers, all of which connect back to these sequences.

Sources and methodology

Figurate numbers are classical; these are the references.

• OEIS A000217 — the triangular numbers, with their many appearances catalogued
• American Mathematical Society — where figurate numbers and Euler’s pentagonal number theorem are catalogued
• Consecutive Integers Calculator — triangular numbers are sums of consecutive integers from 1
• How we calculate — methodology

Method. Values are computed on arbitrary-precision integers, so the hundred-thousandth tetrahedral number is exact rather than rounded through a double. The triangularity test uses the closed 8N + 1 condition with an integer square root rather than searching, and the suite checks it against a direct search over every value to 2,000 — a closed test that disagreed with the search would be worse than no test. That engine is verified on every change against 99 hand-written assertions, including that the 8N + 1 test agrees with a direct search to 2,000, that triangular and tetrahedral values match running sums, and that every hexagonal number is triangular. The count and the per-case breakdown are published on the formula verification page.

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Read the guide

A triangular number is the sum of consecutive integers from 1 — the Consecutive Integers Calculator handles runs starting anywhere.