The triangular number 
is a figurate number that can be represented in
the form of a triangular grid of points where the first row contains a single element
and each subsequent row contains one more element than the previous one. This is
illustrated above for 
,
, .... The triangular numbers are
therefore 1, 
,
, 
, ..., so for 
, 2, ..., the first few are 1, 3, 6, 10, 15, 21, ... (OEIS
A000217).
More formally, a triangular number is a number obtained by adding all positive integers less than or equal to a given positive integer 
, i.e.,
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is a binomial coefficient. For the identities
and representation counts below, set 
. As a result, the number of distinct wine glass clinks
that can be made among a group of 
people (which is simply 
) is given by the triangular number 
.
The triangular number 
is also the dimension of the
special orthogonal group 
, counting the independent rotation
parameters in 
-dimensional
Euclidean space. There is one coordinate plane
for each pair of coordinate axes, giving 
such planes. Thus three-dimensional rotations
have 3 independent parameters and four-dimensional rotations
have 6.
The triangular number 
is therefore the additive analog of the
factorial 
.
A plot of the first few triangular numbers represented as a sequence of binary bits is shown above. The top portion shows 
to 
, and the bottom shows the next 510 values.
The odd triangular numbers are given by 1, 3, 15, 21, 45, 55, ... (OEIS A014493), while the positive even triangular numbers are 6, 10, 28, 36, 66, 78, ... (OEIS A014494).
gives the number and arrangement of the tetractys (which
is also the arrangement of bowling pins), while 
gives the number and arrangement
of balls in billiards. Triangular numbers satisfy the
recurrence relation
 |
(4)
|
as well as
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
In addition, the triangle numbers can be related to the square numbers by
 |  |  |
(9)
|
 |  |  |
(10)
|
(Conway and Guy 1996), as illustrated above (Wells 1991, p. 198).
The triangular numbers have the ordinary generating function 
for 
,
 |  |  |
(11)
|
 |  |  |
(12)
|
and shifted exponential generating function 
,
 |  |  |
(13)
|
 |  |  |
(14)
|
 |  |  |
(15)
|
(Sloane and Plouffe 1995, p. 9).
Every other triangular number 
is a hexagonal number,
with
 |
(16)
|
In addition, every pentagonal number is 1/3 of a triangular number, with
 |
(17)
|
The sum of consecutive triangular numbers is a square number, since
 |  |  |
(18)
|
 |  | ![1/2r[(r+1)+(r-1)]](/images/equations/TriangularNumber/Inline75.svg) |
(19)
|
 |  |  |
(20)
|
Interesting identities involving triangular, square, and cubic numbers are
 |  |  |
(21)
|
 |  |  |
(22)
|
 |  |  |
(23)
|
 |  |  |
(24)
|
 |  |  |
(25)
|
Triangular numbers also unexpectedly appear in integrals involving the absolute value of the form
 |
(26)
|
All even perfect numbers are triangular 
with prime 
. Furthermore, every even perfect number 
is of the form
 |
(27)
|
where 
is a triangular number with 
(Eaton 1995, 1996). Therefore, the nested expression
 |
(28)
|
generates triangular numbers for any 
. An integer 
is a triangular number iff 
is a square number 
.
The positive numbers 1, 36, 1225, 41616, 1413721, 48024900, ... (OEIS A001110) are square triangular numbers, i.e., numbers
which are simultaneously triangular and square (Pietenpol
1962). The corresponding square roots are 1, 6, 35, 204, 1189, 6930, ... (OEIS A001109), and the indices of the corresponding
triangular numbers 
are 
,
8, 49, 288, 1681, ... (OEIS A001108).
Numbers which are simultaneously triangular and tetrahedral satisfy the binomial coefficient equation
 |
(29)
|
the only solutions with positive integer indices are
 |  |  |
(30)
|
 |  |  |
(31)
|
 |  |  |
(32)
|
 |  |  |
(33)
|
 |  |  |
(34)
|
(Guy 1994, p. 147).
The following table gives triangular numbers 
having prime indices 
.
| class | Sloane | sequence |
 with prime indices | A034953 | 3, 6, 15, 28, 66, 91, 153, 190, 276, 435, 496, ... |
odd 
with prime indices | A034954 | 3, 15, 91, 153, 435, 703, 861, 1431, 1891, 2701, ... |
even 
with prime indices | A034955 | 6, 28, 66, 190, 276, 496, 946, 1128, 1770, 2278, ... |
The smallest of two integers for which 
is four times a triangular number is 5, as determined
by Cesàro in 1886 (Le Lionnais 1983, p. 56). The only Fibonacci
numbers which are triangular are 1, 3, 21, and 55 (Ming 1989), and the only Pell number which is triangular is 1 (McDaniel 1996).
The beast number 666 is triangular, since
 |
(35)
|
In fact, it is the largest repdigit triangular number (Ballew and Weger 1975-76; Hercher and Fegert 2025, p. 2).
The positive divisors of 
are all of the form 
, those of 
are all of the form 
, and those of 
are all of the form 
; that is, they end in the decimal digit 1 or 9.
Fermat's polygonal number theorem states that every positive integer is a sum of
at most three triangular numbers, four square
numbers, five pentagonal numbers, and 
-polygonal numbers. Gauss
proved the triangular case (Wells 1986, p. 47), and noted the event in his diary
on July 10, 1796, with the notation
 |
(36)
|
This case is equivalent to the statement that, for every nonnegative integer 
,
the number 
is a sum of three odd squares
(Duke 1997, pp. 190-191). Dirichlet derived the number of ordered triples of
nonnegative integer indices 
satisfying 
(Duke 1997, p. 191). The result is particularly
simple for a prime of the
form 
,
in which case it is the number of squares mod 
minus the number of nonsquares mod 
in the interval from 1 to
(Deligne 1974; Duke 1997, p. 191).
The only positive triangular numbers which are the product of three consecutive integers are 6, 120, 210, 990, 185136, 258474216 (OEIS A001219; Tzanakis and de Weger 1989, p. 119; Guy 1994, p. 148).
