Triangular Number


The triangular number T_n 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 T_1=1, T_2=3, .... The triangular numbers are therefore 1, 1+2, 1+2+3, 1+2+3+4, ..., so for n=1, 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 n, i.e.,

=(n+1; 2),

where (n; k) is a binomial coefficient. As a result, the number of distinct wine glass clinks that can be made among a group of n people (which is simply (n; 2)) is given by the triangular number T_(n-1).

The triangular number T_n=n+(n-1)+...+2+1 is therefore the additive analog of the factorial n!=n·(n-1)...2·1.

Binary representation of the triangular numbers

A plot of the first few triangular numbers represented as a sequence of binary bits is shown above. The top portion shows T_1 to T_(255), 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).

T_4=10 gives the number and arrangement of the tetractys (which is also the arrangement of bowling pins), while T_5=15 gives the number and arrangement of balls in billiards. Triangular numbers satisfy the recurrence relation


as well as


In addition, the triangle numbers can be related to the square numbers by


(Conway and Guy 1996), as illustrated above (Wells 1991, p. 198).

The triangular numbers have the ordinary generating function


and exponential generating function


(Sloane and Plouffe 1995, p. 9).

Every other triangular number T_n is a hexagonal number, with


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


The sum of consecutive triangular numbers is a square number, since


Interesting identities involving triangular, square, and cubic numbers are


Triangular numbers also unexpectedly appear in integrals involving the absolute value of the form


All even perfect numbers are triangular T_p with prime p. Furthermore, every even perfect number P>6 is of the form


where T_n is a triangular number with n=8j+2 (Eaton 1995, 1996). Therefore, the nested expression


generates triangular numbers for any T_n. An integer k is a triangular number iff 8k+1 is a square number >1.

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 T_n are n=1, 8, 49, 288, 1681, ... (OEIS A001108).

Numbers which are simultaneously triangular and tetrahedral satisfy the binomial coefficient equation

 T_n=(n+1; 2)=(m+2; 3)=Te_m,

the only solutions of which are


(Guy 1994, p. 147).

The following table gives triangular numbers T_p having prime indices p.

T_n with prime indicesA0349533, 6, 15, 28, 66, 91, 153, 190, 276, 435, 496, ...
odd T_n with prime indicesA0349543, 15, 91, 153, 435, 703, 861, 1431, 1891, 2701, ...
even T_n with prime indicesA0349556, 28, 66, 190, 276, 496, 946, 1128, 1770, 2278, ...

The smallest of two integers for which n^3-13 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


In fact, it is the largest repdigit triangular number (Bellew and Weger 1975-76).

The positive divisors of 4T(n)+1 are all of the form 4k+1, those of 6T(n)+1 are all of the form 6k+1, and those of 10T(n)+1 are all of the form 10k+/-1; 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 n n-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

 **EUpsilonRHKA    num=Delta+Delta+Delta.

This case is equivalent to the statement that every number of the form 8m+3 is a sum of three odd squares (Duke 1997). Dirichlet derived the number of ways in which an integer m can be expressed as the sum of three triangular numbers (Duke 1997). The result is particularly simple for a prime of the form 8m+3, in which case it is the number of squares mod 8m+3 minus the number of nonsquares mod 8m+3 in the interval from 1 to 4m+1 (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).

See also

Bowling, Cubic Triangular Number, Figurate Number, Heptagonal Triangular Number, Octagonal Triangular Number, Pentagonal Triangular Number, Pronic Number, Square Triangular Number, Tetractys

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Triangular Number

Cite this as:

Weisstein, Eric W. "Triangular Number." From MathWorld--A Wolfram Web Resource.

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