The triangular number is afigurate 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, ... (OEISA000217).

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 abinomial 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 thefactorial.

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, ... (OEISA014493),while the even triangular numbers are 6, 10, 28, 36, 66, 78, ... (OEISA014494).

gives the number and arrangement of thetetractys (which is also the arrangement ofbowling pins), while gives the number and arrangement of balls inbilliards. Triangular numbers satisfy therecurrence 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 ordinarygeneratingfunction

			(11)
			(12)

andexponential generating function

			(13)
			(14)
			(15)

(Sloane and Plouffe 1995, p. 9).

Every other triangular number is ahexagonal number, with

(16)

In addition, everypentagonal number is 1/3of a triangular number, with

(17)

The sum of consecutive triangular numbers is asquarenumber, since

			(18)
			(19)
			(20)

Interesting identities involving triangular,square,andcubic numbers are

			(21)
			(22)
			(23)
			(24)
			(25)

Triangular numbers also unexpectedly appear in integrals involving theabsolutevalueof the form

(26)

Allevenperfect numbers are triangular withprime. Furthermore, everyevenperfect number isof the form

(27)

where is a triangular number with (Eaton 1995, 1996). Therefore, the nested expression

(28)

generates triangular numbers for any. Aninteger is a triangular numberiff is asquare number.

The numbers 1, 36, 1225, 41616, 1413721, 48024900, ... (OEISA001110) aresquare triangular numbers, i.e., numbers which are simultaneously triangular andsquare (Pietenpol 1962). The corresponding square roots are 1, 6, 35, 204, 1189, 6930, ... (OEISA001109), and the indices of the corresponding triangular numbers are, 8, 49, 288, 1681, ... (OEISA001108).

Numbers which are simultaneously triangular andtetrahedralsatisfy thebinomial 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 twointegers for which is four times a triangular number is 5, as determined by Cesàro in 1886 (Le Lionnais 1983, p. 56). The onlyFibonacci numbers which are triangular are 1, 3, 21, and 55 (Ming 1989), and the onlyPell number which is triangular is 1 (McDaniel 1996). Thebeast number 666 is triangular, since

(34)

In fact, it is the largestrepdigit 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 everypositive integer is a sum of atmost three triangular numbers, foursquare numbers, fivepentagonal 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 numberof the form is a sum of threeoddsquares (Duke 1997). Dirichlet derived the number of ways in which aninteger can be expressed as the sum of three triangular numbers (Duke 1997). The result is particularly simple for aprimeof the form, in which case it is the number of squares mod minus the number of nonsquares mod in theinterval from 1 to (Deligne 1973, Duke 1997).

The only triangular numbers which are theproduct of three consecutiveintegers are 6, 120, 210, 990, 185136, 258474216 (OEISA001219; Guy 1994, p. 148).

See also

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More things to try:

• triangular number
• 55385th triangular number = 31977th pentagonal number = 27693th hexagonal number
• 165th pentagonal number = 285th triangular number = 143th hexagonal number

References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Triangular Number."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/TriangularNumber.html

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