Triangular number

A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The nth triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666...

The first six triangular numbers (not starting with T₀)

Triangular Numbers Plot

(sequence A000217 in the OEIS)

Formula

Derivation of triangular numbers from a left-justified Pascal's triangle.

  Natural numbers

  Triangular numbers

  Tetrahedral numbers

  Pentatope numbers

  5-simplex numbers

  6-simplex numbers

  7-simplex numbers

The triangular numbers are given by the following explicit formulas:

${\displaystyle \displaystyle {\begin{aligned}T_{n}&=\sum _{k=1}^{n}k=1+2+\dotsb +n\\&={\frac {n^{2}+n{\vphantom {(n+1)}}}{2}}={\frac {n(n+1)}{2}}\\&={n+1 \choose 2}\end{aligned}}}$

where ${\displaystyle \textstyle {n+1 \choose 2}}$ is notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as "n plus one choose two".

The fact that the ${\displaystyle n}$th triangular number equals ${\displaystyle n(n+1)/2}$ can be illustrated using a visual proof.^[1] For every triangular number ${\displaystyle T_{n}}$, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions ${\displaystyle n\times (n+1)}$, which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or: ${\displaystyle T_{n}={\frac {n(n+1)}{2}}}$. The example ${\displaystyle T_{4}}$ follows:

${\displaystyle 2T_{4}=4(4+1)=20}$ (green plus yellow) implies that ${\displaystyle T_{4}={\frac {4(4+1)}{2}}=10}$ (green).   

This formula can be proven formally using mathematical induction.^[2] It is clearly true for ${\displaystyle 1}$:

$${\displaystyle T_{1}=\sum _{k=1}^{1}k={\frac {1(1+1)}{2}}={\frac {2}{2}}=1.}$$

Now assume that, for some natural number ${\displaystyle m}$, ${\displaystyle T_{m}=\sum _{k=1}^{m}k={\frac {m(m+1)}{2}}}$. Adding ${\displaystyle m+1}$ to this yields

$${\displaystyle {\begin{aligned}\sum _{k=1}^{m}k+(m+1)&={\frac {m(m+1)}{2}}+m+1\\&={\frac {m(m+1)+2m+2}{2}}\\&={\frac {m^{2}+m+2m+2}{2}}\\&={\frac {m^{2}+3m+2}{2}}\\&={\frac {(m+1)(m+2)}{2}},\end{aligned}}}$$

so if the formula is true for ${\displaystyle m}$, it is true for ${\displaystyle m+1}$. Since it is clearly true for ${\displaystyle 1}$, it is therefore true for ${\displaystyle 2}$, ${\displaystyle 3}$, and ultimately all natural numbers ${\displaystyle n}$ by induction.

The German mathematician and scientist, Carl Friedrich Gauss, is said to have found this relationship in his early youth, by multiplying ⁠n/2⁠ pairs of numbers in the sum by the values of each pair n + 1.^[3] However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans in the 5th century BC.^[4] The two formulas were described by the Irish monk Dicuil in about 816 in his Computus.^[5] An English translation of Dicuil's account is available.^[6]

Proof without words that the number of possible handshakes between n people is the (n−1)th triangular number

The triangular number T_n solves the handshake problem of counting the number of handshakes if each person in a room with n + 1 people shakes hands once with each person. In other words, the solution to the handshake problem of n people is T_n−1.^[7] The function T is the additive analog of the factorial function, which is the products of integers from 1 to n.

This same function was coined as the "Termial function"^[8] by Donald Knuth's The Art of Computer Programming and denoted n? (analog for the factorial notation n!)

For example, 10 termial is equivalent to:

$${\displaystyle 10?=1+2+3+4+5+6+7+8+9+10=55}$$

which of course, corresponds to the tenth triangular number.

The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a recurrence relation: $${\displaystyle L_{n}=3T_{n-1}=3{n \choose 2};~~~L_{n}=L_{n-1}+3(n-1),~L_{1}=0.}$$

In the limit, the ratio between the two numbers, dots and line segments is $${\displaystyle \lim _{n\to \infty }{\frac {T_{n}}{L_{n}}}={\frac {1}{3}}.}$$

Relations to other figurate numbers

Triangular numbers have a wide variety of relations to other figurate numbers.

Most simply, the sum of two consecutive triangular numbers is a square number, since:^[9][10]

${\displaystyle T_{n-1}+T_{n}}$

${\displaystyle ={\frac {1}{2}}\,n(n-1)+{\frac {1}{2}}\,n(n+1)}$

${\displaystyle ={\frac {1}{2}}\,n{\Bigl (}(n-1)+(n+1){\Bigr )}}$

${\displaystyle =n^{2}}$

with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum): $${\displaystyle T_{n}+T_{n-1}=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {\left(n-1\right)^{2}}{2}}+{\frac {n-1{\vphantom {\left(n-1\right)^{2}}}}{2}}\right)=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {n^{2}}{2}}-{\frac {n}{2}}\right)=n^{2}=(T_{n}-T_{n-1})^{2}.}$$

This property, colloquially known as the theorem of Theon of Smyrna,^[11] is visually demonstrated in the following sum, which represents ${\displaystyle T_{4}+T_{5}=5^{2}}$ as digit sums:

${\displaystyle {\begin{array}{ccccccc}&4&3&2&1&\\+&1&2&3&4&5\\\hline &5&5&5&5&5\end{array}}}$

This fact can also be demonstrated graphically by positioning the triangles in opposite directions to create a square:

6 + 10 = 16         10 + 15 = 25    

The double of a triangular number, as in the visual proof from the above section § Formula, is called a pronic number.

There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36, 1225. Some of them can be generated by a simple recursive formula: $${\displaystyle S_{n+1}=4S_{n}\left(8S_{n}+1\right)}$$ with ${\displaystyle S_{1}=1.}$

All square triangular numbers are found from the recursion $${\displaystyle S_{n}=34S_{n-1}-S_{n-2}+2}$$ with ${\displaystyle S_{0}=0}$ and ${\displaystyle S_{1}=1.}$

A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. This shows that the square of the nth triangular number is equal to the sum of the first n cube numbers.

Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n. This can also be expressed as $${\displaystyle \sum _{k=1}^{n}k^{3}=\left(\sum _{k=1}^{n}k\right)^{2}.}$$

The sum of the first n triangular numbers is the nth tetrahedral number: $${\displaystyle \sum _{k=1}^{n}T_{k}=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}={\frac {n(n+1)(n+2)}{6}}.}$$

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 $${\displaystyle Ck_{n}=kT_{n-1}+1}$$

where T is a triangular number.

The positive difference of two triangular numbers is a trapezoidal number.

The pattern found for triangular numbers ${\displaystyle \sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{2}+1}{2}}}$ and for tetrahedral numbers ${\displaystyle \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{3}+2}{3}},}$ which uses binomial coefficients, can be generalized. This leads to the formula:^[12] $${\displaystyle \sum _{n_{k-1}=1}^{n_{k}}\sum _{n_{k-2}=1}^{n_{k-1}}\dots \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{k}+k-1}{k}}}$$

The fourth triangular number equals the third tetrahedral number as the nth k-simplex number equals the kth n-simplex number due to the symmetry of Pascal's triangle, and its diagonals being simplex numbers; similarly, the fifth triangular number (15) equals the third pentatope number, and so forth

Other properties

Triangular numbers correspond to the first-degree case of Faulhaber's formula.

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Proof without words that all hexagonal numbers are odd-sided triangular numbers

Alternating triangular numbers (1, 6, 15, 28, ...) are also hexagonal numbers.

Every even perfect number is triangular (as well as hexagonal), given by the formula $${\displaystyle M_{p}2^{p-1}={\frac {M_{p}(M_{p}+1)}{2}}=T_{M_{p}}}$$ where M_p is a Mersenne prime. No odd perfect numbers are known; hence, all known perfect numbers are triangular.

For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.

The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.

In base 10, the digital root of a nonzero 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 9:

0 = 9 × 0

1 = 9 × 0 + 1

3 = 9 × 0 + 3

6 = 9 × 0 + 6

10 = 9 × 1 + 1

15 = 9 × 1 + 6

21 = 9 × 2 + 3

28 = 9 × 3 + 1

36 = 9 × 4

45 = 9 × 5

55 = 9 × 6 + 1

66 = 9 × 7 + 3

78 = 9 × 8 + 6

91 = 9 × 10 + 1

...

The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9".

The converse 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.

If x is a triangular number, then ax + b is also a triangular number, given a is an odd square and b = ⁠a − 1/8⁠. Note that b will always be a triangular number, because 8T_n + 1 = (2n + 1)², which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for b given a is an odd square is the inverse of this operation. The first several pairs of this form (not counting 1x + 0) are: 9x + 1, 25x + 3, 49x + 6, 81x + 10, 121x + 15, 169x + 21, ... etc. Given x is equal to T_n, these formulas yield T_{3n + 1}, T_{5n + 2}, T_{7n + 3}, T_{9n + 4}, and so on.

The sum of the reciprocals of all the nonzero triangular numbers is $${\displaystyle \sum _{n=1}^{\infty }{1 \over {{n^{2}+n} \over 2}}=2\sum _{n=1}^{\infty }{1 \over {n^{2}+n}}=2.}$$

This can be shown by using the basic sum of a telescoping series: $${\displaystyle \sum _{n=1}^{\infty }{1 \over {n(n+1)}}=1.}$$

Two other formulas regarding triangular numbers are $${\displaystyle T_{a+b}=T_{a}+T_{b}+ab}$$ and $${\displaystyle T_{ab}=T_{a}T_{b}+T_{a-1}T_{b-1},}$$ both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra.

In 1796, Gauss discovered that every positive integer is representable as a sum of three triangular numbers (possibly including T₀ = 0), writing in his diary his famous words, "ΕΥΡΗΚΑ! num = Δ + Δ + Δ". This theorem does not imply that the triangular numbers are different (as in the case of 20 = 10 + 10 + 0), nor that a solution with exactly three nonzero triangular numbers must exist. This is a special case of the Fermat polygonal number theorem.

The largest triangular number of the form 2^k − 1 is 4095 (see Ramanujan–Nagell equation).

Wacław Franciszek Sierpiński posed the question as to the existence of four distinct triangular numbers in geometric progression. It was conjectured by Polish mathematician Kazimierz Szymiczek to be impossible and was later proven by Fang and Chen in 2007.^[13][14]

Formulas involving expressing an integer as the sum of triangular numbers are connected to theta functions, in particular the Ramanujan theta function.^[15][16]

Applications

The maximum number of pieces, p obtainable with n straight cuts is the n-th triangular number plus one, forming the lazy caterer's sequence (OEIS A000124)

A fully connected network of n computing devices requires the presence of T_{n − 1} cables or other connections; this is equivalent to the handshake problem mentioned above.

In a tournament format that uses a round-robin group stage, the number of matches that need to be played between n teams is equal to the triangular number T_{n − 1}. For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems.

One way of calculating the depreciation of an asset is the sum-of-years' digits method, which involves finding T_n, where n is the length in years of the asset's useful life. Each year, the item loses (b − s) × ⁠n − y/T_n⁠, where b is the item's beginning value (in units of currency), s is its final salvage value, n is the total number of years the item is usable, and y the current year in the depreciation schedule. Under this method, an item with a usable life of n = 4 years would lose ⁠4/10⁠ of its "losable" value in the first year, ⁠3/10⁠ in the second, ⁠2/10⁠ in the third, and ⁠1/10⁠ in the fourth, accumulating a total depreciation of ⁠10/10⁠ (the whole) of the losable value.

Board game designers Geoffrey Engelstein and Isaac Shalev describe triangular numbers as having achieved "nearly the status of a mantra or koan among game designers", describing them as "deeply intuitive" and "featured in an enormous number of games, [proving] incredibly versatile at providing escalating rewards for larger sets without overly incentivizing specialization to the exclusion of all other strategies".^[17]

Relationship between the maximum number of pips on an end of a domino and the number of dominoes in its set
(values in bold are common)

Max. pips	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22
Tn	1	3	6	10	15	21	28	36	45	55	66	78	91	105	120	136	153	161	190	210	231	253

Triangular roots and tests for triangular numbers

By analogy with the square root of x, one can define the (positive) triangular root of x as the number n such that T_n = x:^[18] $${\displaystyle n={\frac {{\sqrt {8x+1}}-1}{2}}}$$

which follows immediately from the quadratic formula. So an integer x is triangular if and only if 8x + 1 is a square. Equivalently, if the positive triangular root n of x is an integer, then x is the nth triangular number.^[18]

Alternative name

As stated, an alternative name proposed by Donald Knuth, by analogy to factorials, is "termial", with the notation n? for the nth triangular number.^[19] However, although some other sources use this name and notation,^[20] they are not in wide use.

See also

• 1 + 2 + 3 + 4 + ⋯
• Doubly triangular number, a triangular number whose position in the sequence of triangular numbers is also a triangular number
• Tetractys, an arrangement of ten points in a triangle, important in Pythagoreanism

References

1. "Triangular Number Sequence". Math Is Fun.
2. Spivak, Michael (2008). Calculus (4th ed.). Houston, Texas: Publish or Perish. pp. 21–22. ISBN 978-0-914098-91-1.
3. Hayes, Brian. "Gauss's Day of Reckoning". American Scientist. Computing Science. Archived from the original on 2015-04-02. Retrieved 2014-04-16.
4. Eves, Howard. "Webpage cites AN INTRODUCTION TO THE HISTORY OF MATHEMATICS". Mathcentral. Retrieved 28 March 2015.
5. Esposito, M. An unpublished astronomical treatise by the Irish monk Dicuil. Proceedings of the Royal Irish Academy, XXXVI C. Dublin, 1907, 378-446.
6. Ross, H.E. & Knott, B.I."Dicuil (9th century) on triangular and square numbers." British Journal for the History of Mathematics, 2019,34 (2), 79-94. https://doi.org/10.1080/26375451.2019.1598687.
7. "The Handshake Problem | National Association of Math Circles". MathCircles.org. Archived from the original on 10 March 2016. Retrieved 12 January 2022.
8. Knuth, Donald. The Art of Computer Programming. Vol. 1 (3rd ed.). p. 48.
9. Beldon, Tom; Gardiner, Tony (2002). "Triangular Numbers and Perfect Squares". The Mathematical Gazette. 86 (507): 423–431. doi:10.2307/3621134. JSTOR 3621134. Retrieved 25 April 2024.
10. Eric W. Weisstein. "Triangular Number". Wolfram MathWorld. Retrieved 2024-04-14. See equations 18 - 20.
11. Shell-Gellasch, Amy; Thoo, John (October 15, 2015). Algebra in Context: Introductory Algebra from Origins to Applications. Johns Hopkins University Press. p. 210. doi:10.1353/book.49475. ISBN 9781421417288.
12. Baumann, Michael Heinrich (2018-12-12). "Die k-dimensionale Champagnerpyramide" (PDF). Mathematische Semesterberichte (in German). 66: 89–100. doi:10.1007/s00591-018-00236-x. ISSN 1432-1815. S2CID 125426184.
13. Chen, Fang: Triangular numbers in geometric progression
14. Fang: Nonexistence of a geometric progression that contains four triangular numbers
15. Liu, Zhi-Guo (2003-12-01). "An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers". The Ramanujan Journal. 7 (4): 407–434. doi:10.1023/B:RAMA.0000012425.42327.ae. ISSN 1382-4090. S2CID 122221070.
16. Sun, Zhi-Hong (2016-01-24). "Ramanujan's theta functions and sums of triangular numbers". arXiv:1601.06378 [math.NT].
17. Engelstein, Geoffrey; Shalev, Isaac (2019-06-25). Building Blocks of Tabletop Game Design. doi:10.1201/9780429430701. ISBN 978-0-429-43070-1. S2CID 198342061.
18. Euler, Leonhard; Lagrange, Joseph Louis (1810), Elements of Algebra, vol. 1 (2nd ed.), J. Johnson and Co., pp. 332–335
19. Donald E. Knuth (1997). The Art of Computer Programming: Volume 1: Fundamental Algorithms. 3rd Ed. Addison Wesley Longman, U.S.A. p. 48.
20. Stone, John David (2018), Algorithms for Functional Programming, Springer, p. 282, doi:10.1007/978-3-662-57970-1, ISBN 978-3-662-57968-8, S2CID 53079729

External links

• "Arithmetic series", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Triangular numbers at cut-the-knot
• There exist triangular numbers that are also square at cut-the-knot
• Weisstein, Eric W. "Triangular Number". MathWorld.
• Hypertetrahedral Polytopic Roots by Rob Hubbard, including the generalisation to triangular cube roots, some higher dimensions, and some approximate formulas