A triangular number is the sum of the n natural numbers from 1 to n. Triangular numbers are so called because they describe numbers of objects that can be arranged in a triangle. The nth triangular number is given by In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

As shown in the rightmost term of this formula, every triangular number is a binomial coefficient: the nth triangular is the number of distinct pairs to be selected from n + 1 objects. In this form it solves the 'handshake problem' of counting the number of handshakes if each person in a room shakes hands once with each other person. In mathematics, particularly in combinatorics, a binomial coefficient is a coefficient of any of the terms in the expansion of the binomial (x+1)n. ...

The sequence of triangular numbers (sequence A000217 in OEIS) for n = 1, 2, 3... is: The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

This article is about the number one. ... Look up three in Wiktionary, the free dictionary. ... Look up six in Wiktionary, the free dictionary. ... This article is about the number 10. ... 15 (fifteen) is the natural number following 14 and preceding 16. ... 21 (twenty-one) is the natural number following 20 and preceding 22. ... 28 (twenty-eight) is the natural number following 27 and preceding 29. ... 36 (thirty-six) is the natural number following 35 and preceding 37. ... 45 (forty-five) is the natural number following 43 and followed by 47. ... 55 is the natural number following 54 and preceding 56. ...

Relations to other figurate numbers

Triangular numbers have a wide variety of relations to other figurate numbers. A figurate number is a number that can be represented as a regular and discrete geometric pattern (e. ...

Most simply, the sum of two consecutive triangular numbers is a square number. Algebraically, In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ...

Alternatively, the same fact can be demonstrated graphically:

16

25

There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36. Some of them can be generated by a simple recursive formula: Image File history File links The square number 16 as a sum of two triangular numbers. ... Image File history File links The square number 25 as a sum of two triangular numbers. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ...

with S₁ = 1

All square triangular numbers are found from the recursion A square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. ...

S_n = 34S_{n − 1} − S_{n − 2} + 2 with S₀ = 0 and S₁ = 1

Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n. Visual demonstration that the square of a triangular number equals a sum of cubes. ...

The sum of the n first triangular numbers is the nth tetrahedral number, A pyramid with side length 5 contains 35 spheres. ...

More generally, the difference between the nth m-gonal number and the nth (m + 1)-gonal number is the (n - 1)th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number: the nth centered k-gonal number is obtained by the formula In mathematics, a polygonal number is a number that can be arranged as a regular polygon. ... A heptagonal number is a figurate number that represents a heptagon. ... A hexagonal number is a figurate number that represents a hexagon. ... A hexagonal number is a figurate number that represents a hexagon. ... Centred numbers are class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. ...

Ck_n = kT_{n − 1} + 1

where T is a triangular number.

Other properties

Every even perfect number is triangular, and no odd perfect numbers are known, hence all known perfect numbers are triangular. In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ...

In base 10, the digital root of a triangular number is always 1, 3, 6 or 9. Hence every triangular number is either divisible by three or has a remainder of 1 when divided by nine: Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus... It has been suggested that Repeated digital sum be merged into this article or section. ...

6 = 3×2,

10 = 9×1+1,

15 = 3×5,

21 = 3×7,

28 = 9×3+1,

...

The inverse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.

The sum of the reciprocals of all the triangular numbers is: Look up reciprocal in Wiktionary, the free dictionary. ...

This can be shown by using the basic sum of a telescoping series: In mathematics, telescoping series is an informal expression referring to a series whose sum can be found by exploiting the circumstance that nearly every term cancels with a succeeding or preceding term. ...

Two other interesting formulas regarding triangular numbers are:

T_{a + b} = T_a + T_b + ab

and

T_ab = T_aT_b + T_{a − 1}T_{b − 1},

both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra.

In 1796, German mathematician and scientist Carl Friedrich Gauss discovered that every positive integer is representable as a sum of at most three triangular numbers, writing in his diary his famous words, "Heureka! num= Δ + Δ + Δ." Note (as in the case of 20=10+10) that this theorem does not imply that the triangular numbers are different, nor that a solution with three nonzero triangular numbers must exist. This is a special case of Fermat's Polygonal Number Theorem. Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Every positive integer is a sum of at most -polygonal numbers. ...

Tests for triangular numbers

One can efficiently test whether a positive integer x is a triangular number by computing

If n is an integer, then x is the nth triangular number. If n is not an integer, then x is not triangular.

External links

• Triangular numbers at cut-the-knot
• There exist triangular numbers that are also square at cut-the-knot
• Mathworld