Triangular Number

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)
			(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

			(11)
			(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)
			(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).

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