What are triangular numbers?

Triangular numbers are the integers that count dots arranged in equilateral triangles: 1, 3, 6, 10, 15, 21, 28, 36, and so on. The n-th triangular number equals the sum of the first n positive integers, given by the closed form T sub n equals n times the quantity n plus 1, divided by 2.

What is the triangular number formula?

The closed form is T sub n equals n times the quantity n plus 1, divided by 2. Equivalently, T sub n equals the sum from k equals 1 to n of k. The formula gives an exact integer result for every positive integer n because either n or n plus 1 is even.

How do you test whether a number is triangular?

A positive integer m is triangular if and only if 8m plus 1 is a perfect square. If the square root of 8m plus 1 is the integer s, then m equals T sub n where n equals the quantity s minus 1, divided by 2. This test is exact and constant-time.

What is the sum of the first n triangular numbers?

The sum of the first n triangular numbers is the n-th tetrahedral number: T sub 1 plus T sub 2 plus T sub n equals n times the quantity n plus 1 times the quantity n plus 2, divided by 6. Tetrahedral numbers count dots in a triangular pyramid the way triangular numbers count dots in a triangle.

Where do triangular numbers appear in mathematics?

Triangular numbers appear in combinatorics (the number of unordered pairs from n plus 1 objects equals T sub n), in algebra (the number of handshakes among n plus 1 people), in geometry (lattice points in a triangle), and as a row of Pascal's triangle. They also appear in the partial sums of arithmetic series and in the Gauss summation identity.

Triangular Numbers Calculator — Browse, Test, Lookup

Listing the First N Terms

DemoTen terms, thirty terms, the cap, one term

Step 0 of 4

The explorer opens on the List first N tab, one of four tabs across the top of the card; the small ? beside each tab shows what the mode does. Type how many terms you want in the Show first box and click Update or press Enter. Three cards answer at once: Showing gives the count, Largest the last term with its index, and Sum the total of the listed terms, which for triangular numbers is the tetrahedral number

N(N+1)(N+2)/6

. Under the cards every term is laid out as a tile with its index above its value, so

T_1 = 1

to

T_N

can be read in order.

The box accepts

1

to

200

:

10

gives

T_{10} = 55

with the sum

220

,

30

gives

T_{30} = 465

with the sum

4{,}960

, and

300

is capped at

200

terms, ending at

T_{200} = 20{,}100

with the sum

1{,}353{,}400

; a single term gives

T_1 = 1

. Large values are printed with thousands separators. A count of 0, a decimal or an empty box is refused with "Enter a positive integer." There is no Reset button: type a new count and click Update again. The panel on the right, List first N, states the closed form the tiles come from, and its walkthrough repeats the last term and the sum with your own

N

. Why the terms are the sums

1 + 2 + \cdots + n

is explained in what is a triangular number.

Listing a Range of Indexes

DemoThe range tab, 10 to 20, two bad ranges

Step 0 of 4

Click the List a – b tab to list the terms between two positions instead of starting from

1

. Type the From index and the To index and click List range or press Enter. The cards give the Count

b - a + 1

, the Smallest term

T_a

and the Largest term

T_b

, and the tiles below list every term in between with its index. The tab opens with

5

to

15

, eleven terms from

T_5 = 15

to

T_{15} = 120

; indexes

10

to

20

give eleven terms from

55

to

210

.

Both boxes must hold positive integers, the first no larger than the second, and the range may hold at most

200

terms; otherwise a red box explains: "Enter two positive integer indexes", "From-index must be less than or equal to to-index", or "Range too wide — cap is 200 terms". The panel on the right, List a – b, explains the mode, and its walkthrough names the smallest and largest term of your range. The formula each tile uses is proved in the closed form and why it works.

Testing a Number

DemoThe test tab, 100, 5050, 0

Step 0 of 4

Click the Test a number tab to ask whether a given positive integer is triangular. Type it in the Test number box and click Check or press Enter. A green box with a tick answers yes and names the index, as for the opening value

120

, which is

T_{15}

; a red box with a cross answers no, as for

100

, and the walkthrough on the right then shows the two nearest triangular numbers,

T_{13} = 91

and

T_{14} = 105

. A large value is handled the same way:

5{,}050

is

T_{100}

.

The box takes positive integers only;

0

, a negative number or a decimal is refused with "Enter a positive integer". The test behind the answer is exact: the explorer checks whether

8m + 1

is a perfect square and, when it is, reads the index off its square root, so no list has to be searched. The panel on the right, Test a number, states this rule, and its walkthrough carries out the arithmetic for your number. The rule itself is derived in the membership test.

Looking Up a Term by Index

DemoThe lookup tab, n = 100, n = 1000, n = 0

Step 0 of 4

Click the Lookup by index tab to compute one term directly. Type the position in the Index n box and click Get term or press Enter. Two cards answer: Index repeats

n

and Value gives

T_n

with its index as a subscript. The tab opens with

n = 42

, which gives

T_{42} = 903

;

n = 100

gives

5{,}050

and

n = 1{,}000

gives

500{,}500

. The box takes positive integers only;

0

is refused with "Enter a positive integer index".

The panel on the right has two parts in every mode. How it works names the rule the mode uses, here the closed form

T_n = n(n+1)/2

, and links to the section of this page that explains it; Walkthrough applies the rule to the numbers you typed, so for the lookup it shows

n(n+1)/2

with your

n

filled in and the product worked out. Both change as soon as you switch tabs or click a button. The closed form and its two proofs are in the closed form and why it works.

What is a Triangular Number?

A triangular number is the count of dots needed to form an equilateral triangle of side length

n

. Starting with one dot at the top and adding one more dot to each successive row:

T_1 = 1, \quad T_2 = 3, \quad T_3 = 6, \quad T_4 = 10, \quad T_5 = 15, \quad \ldots

The

n

-th triangular number

T_n

equals the sum of the first

n

positive integers:

T_n = 1 + 2 + 3 + \cdots + n = \sum_{k=1}^{n} k

This sum has a famous closed form, sometimes called the Gauss formula:

T_n = \frac{n(n+1)}{2}

The pairing trick that produces the formula: write the sum forward and backward, add term by term to get

n

pairs of

n + 1

, divide by 2. Carl Friedrich Gauss is said to have rediscovered this at age 9. For deeper coverage see triangular number, arithmetic series, and Gauss summation.

The definition is given in definition and the sequence is placed among the others in triangular numbers.

The Closed Form and Why It Works

Two distinct proofs of

T_n = n(n+1)/2

— both worth knowing.

Algebraic (Gauss pairing). Write the sum and its reverse:

\begin{aligned} S &= 1 + 2 + 3 + \cdots + n \\ S &= n + (n-1) + (n-2) + \cdots + 1 \end{aligned}

Adding column by column gives

n

pairs each summing to

n + 1

:

2S = n(n+1) \implies S = \frac{n(n+1)}{2}

Geometric (rectangle). Two copies of a triangular dot pattern can be rotated and fitted together into an

n

by

n + 1

rectangle. The rectangle has

n(n+1)

dots; one triangle has half:

T_n = \frac{n(n+1)}{2}

Why the formula is always an integer. Of

n

and

n + 1

, one is even, so the product is divisible by 2. The formula produces an integer for every positive integer

n

.

The same derivation is written out in derivation of the closed form.

The Membership Test

Given a positive integer

m

, deciding whether it is triangular reduces to a perfect-square check.

The test.

m

is triangular if and only if

8m + 1

is a perfect square. When it is, the index is

n = \frac{\sqrt{8m + 1} - 1}{2}

Why it works. From

T_n = n(n+1)/2 = m

, multiply by 8 and add 1:

4n^2 + 4n + 1 = 8m + 1 \implies (2n+1)^2 = 8m + 1

So

8m + 1

is a perfect square exactly when

m

is triangular, and the square root recovers

n

.

Example. Is 120 triangular? Compute

8 \cdot 120 + 1 = 961

. Then

\sqrt{961} = 31

, an integer. So

n = (31 - 1)/2 = 15

, meaning

120 = T_{15}

. Indeed

15 \cdot 16 / 2 = 120

.

Non-example. Is 100 triangular? Compute

8 \cdot 100 + 1 = 801

. Then

\sqrt{801} \approx 28.30

, not an integer. So 100 is not triangular. The nearest triangular numbers are

T_{13} = 91

and

T_{14} = 105

.

The identities the test rests on are collected in properties.

Properties and Identities

Triangular numbers satisfy many identities that connect them to other figurate numbers and counting problems.

• Recurrence:

T_n = T_{n-1} + n

with

T_1 = 1

. Each triangular number adds one more row to the triangle.
• Binomial coefficient:

T_n = \binom{n+1}{2}

. The

n

-th triangular number counts unordered pairs from

n + 1

objects.
• Sum of consecutive triangulars:

T_n + T_{n-1} = n^2

. Two consecutive triangular numbers add to a perfect square.
• Sum of first $n$ triangulars:

T_1 + T_2 + \cdots + T_n = \frac{n(n+1)(n+2)}{6} = \mathrm{Te}_n

, the

n

-th tetrahedral number.
• Sum of first $n$ cubes:

1^3 + 2^3 + \cdots + n^3 = T_n^2

. The sum of the first

n

cubes equals the square of the

n

-th triangular number.
• Hexagonal connection: every hexagonal number is also triangular:

H_n = T_{2n-1}

.
• Square triangulars: a triangular number that is also a perfect square satisfies a Pell equation; examples are 1, 36, 1225, 41616, infinitely many but sparse.

The recurrence is the subject of recursive form and the other identities of properties.

Common Applications

Triangular numbers appear naturally in counting problems where every pair, edge, or comparison is counted once.

• Handshakes. In a room of

n + 1

people who all shake hands once, the total number of handshakes is

T_n

.
• Pairs from a set. The number of unordered pairs from

n + 1

items is

\binom{n+1}{2} = T_n

.
• Round-robin tournaments. A tournament where each of

n + 1

players plays every other once has

T_n

matches.
• Edges in a complete graph. The complete graph on

n + 1

vertices,

K_{n+1}

, has

T_n

edges.
• Diagonals plus sides. An

(n + 1)

-gon has

T_n

total segments (diagonals plus sides) connecting its vertices.
• Stacked objects. Bowling pins, billiard balls, and any triangularly-stacked array of

n

rows contain

T_n

objects.
• Lazy caterer. The maximum number of pieces from

n

straight cuts of a pizza is

T_n + 1

— the cake number minus a constant.

The counting identity behind these uses is in properties; the sequence's place among the classic ones is triangular numbers.

Triangular Numbers and Pascal's Triangle

Triangular numbers form a diagonal of Pascal's triangle. The third diagonal (counting from 0) reads 1, 3, 6, 10, 15, 21, exactly the triangular numbers.

The identity

T_n = \binom{n+1}{2}

says: the

n

-th triangular number equals the number of ways to choose 2 items from

n + 1

. The connection between binomial coefficients and figurate numbers generalizes: the

k

-th diagonal of Pascal's triangle gives the

k

-dimensional figurate numbers.

• Diagonal 0: all 1's — the trivial sequence.
• Diagonal 1: 1, 2, 3, 4, the natural numbers (1-dimensional simplex counts).
• Diagonal 2: 1, 3, 6, 10, the triangular numbers (2D triangle counts).
• Diagonal 3: 1, 4, 10, 20, the tetrahedral numbers (3D triangle pyramid counts).
• Diagonal 4: 1, 5, 15, 35, the pentatope numbers (4D simplex counts).

This nesting is captured by the hockey-stick identity: summing any diagonal up to a point gives the next diagonal's entry. The sum of the first

n

triangular numbers being

\mathrm{Te}_n

is one instance.

The binomial form of

T_n

is listed in properties; the neighbouring diagonal of squares is square numbers.

Common Mistakes

• Off-by-one in the index.

T_1 = 1

, not

T_0 = 1

. The convention starts at

n = 1

. Some sources start at

n = 0

with

T_0 = 0

; the explorer here uses

T_1 = 1

.

• Confusing triangular with square or other figurate numbers. A triangular number counts dots in a triangle; a square number counts dots in a square; a pentagonal number counts dots in a pentagon. The closed forms differ:

T_n = n(n+1)/2

,

S_n = n^2

,

P_n = n(3n-1)/2

. Numbers can be both (e.g., 1 and 36 are both triangular and square).

• Misapplying the membership test. The test is

8m + 1

a perfect square, not

m + 1

or

4m + 1

. The factor 8 comes from completing the square in

n^2 + n = 2m

.

• Negative or fractional indexes. The closed form

n(n+1)/2

technically extends to all real

n

, but only positive integers are triangular numbers in the standard sense.

T_{1.5}

has no combinatorial meaning.

• Confusing the sum with the count.

T_n

is the sum of the first

n

integers, not the count. The count is

n

itself.

Index conventions and the two ways of defining a sequence are explained in definition and notation and explicit and recursive definitions.

Related Sequences and Concepts

Square Numbers —

S_n = n^2

. Dots arranged in a square.

T_n + T_{n-1} = n^2

connects triangular and square numbers.

Pentagonal Numbers —

P_n = n(3n-1)/2

. The next figurate family. Membership test uses

24m + 1

.

Hexagonal Numbers —

H_n = n(2n-1)

. Every hexagonal number is triangular:

H_n = T_{2n-1}

.

Tetrahedral Numbers —

\mathrm{Te}_n = n(n+1)(n+2)/6

. The sum of the first

n

triangular numbers; the 3D analog.

Square Triangular Numbers — numbers that are both triangular and square: 1, 36, 1225, 41616, ... They satisfy a Pell equation and grow exponentially.

Arithmetic Series —

T_n

is the sum of the first

n

terms of the arithmetic series with

a_1 = 1

,

d = 1

. The general formula

S_n = n(a_1 + a_n)/2

specializes to

T_n

.

Gauss Sum — the classical formula for

1 + 2 + \cdots + n

, equal to

T_n

. The cornerstone of summation techniques.

Pascal's Triangle — row positions

\binom{n+1}{2} = T_n

. Triangular numbers form the third diagonal.

Binomial Coefficient —

T_n = \binom{n+1}{2}

. The number of unordered pairs from

n + 1

objects.

Combinatorics — the discipline. Triangular numbers solve canonical counting problems: pairs, handshakes, edges in complete graphs.

Figurate Numbers — the general family of polygonal and polyhedral counts. Triangular numbers are the 2D, 3-sided case.

Polygonal Numbers — the umbrella term for triangular, square, pentagonal, hexagonal, heptagonal, octagonal, and so on. The

r

-gonal number formula is

P_n^{(r)} = \frac{n[(r-2)n - (r-4)]}{2}

.

The neighbouring calculators are the square numbers calculator, the pentagonal numbers calculator, the tetrahedral numbers calculator and the arithmetic sequence calculator; the lessons are square numbers and arithmetic sequences.