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. 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 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 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)
|
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(6)
|
 |  |  |
(7)
|
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(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
 |  |  |
(11)
|
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(12)
|
and 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/Inline67.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 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 of which are
 |  |  |
(30)
|
 |  |  |
(31)
|
 |  |  |
(32)
|
 |  |  |
(33)
|
(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
 |
(34)
|
In fact, it is the largest repdigit triangular number (Bellew and Weger 1975-76).
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
 |
(35)
|
This case is equivalent to the statement that every number of the form 
is a sum of three odd squares
(Duke 1997). Dirichlet derived the number of ways in which an integer 
can be expressed as the sum of three
triangular numbers (Duke 1997). 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 1973, Duke 1997).
The only triangular numbers which are the product of three consecutive integers are 6, 120, 210, 990, 185136, 258474216 (OEIS A001219; Guy 1994, p. 148).
