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# Triangular numbers

### From OeisWiki

The th **triangular number** is defined as the sum of the first positive integers

where since it is the empty sum of positive integers (giving the additive identity, i.e. 0), and is a binomial coefficient. The th triangular number is thus one half of the th pronic number (or oblong number).

A000217 Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n, n ≥ 0.

- {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, 703, 741, 780, ...}

## Contents |

This was proved by Euler, by the following trick:

1, 2, 3, ..., n-2, n-1, n + n, n-1, n-2, ..., 3, 2, 1 ---------------------------------- n+1, n+1, n+1, ..., n+1, n+1, n+1 thus t_n = n*(n+1)/2.

That proof is sometimes also credited to Carl Friedrich Gauß:

Theorem.The sum of the first positive integers (the th triangular number ) is equal to .

Proof.(Gauß) We can write . Since addition is commutative, we can also write . If we add up these expressions term by term, left to right, we obtain . Each of these parenthesized addends works out to and there are of these addends. Therefore, and dividing both sides by 2 we get as specified by the theorem. □^{[1]}

## Relations

### Relations involving binomial coefficients

Charles Marion <charliemath@optonline.net> suggested the following two relations:

where gives

## Sum of reciprocals

The partial sums of the reciprocals of the **triangular numbers** gives (easily proved by induction)

The sum of the reciprocals of the **triangular numbers** converges to 2

## Representations of natural numbers as a sum of three triangular numbers

Every natural number may be represented, in at least one way, as a sum of three **triangular numbers** (with up to three nonzero triangular numbers).

Representations | Number of representations | |
---|---|---|

0
| { {0, 0, 0} } | 1 |

1
| { {1, 0, 0} } | 1 |

2
| { {1, 1, 0} } | 1 |

3
| { {3, 0, 0}, {1, 1, 1} } | 2 |

4
| { {3, 1, 0} } | 1 |

5
| { {3, 1, 1} } | 1 |

6
| { {6, 0, 0}, {3, 3, 0} } | 2 |

7
| { {6, 1, 0}, {3, 3, 1} } | 2 |

8
| { {6, 1, 1} } | 1 |

9
| { {6, 3, 0}, {3, 3, 3} } | 2 |

10
| { {10, 0, 0}, {6, 3, 1} } | 2 |

11
| { {10, 1, 0} } | 1 |

12
| { {10, 1, 1}, {6, 6, 0}, {6, 3, 3} } | 3 |

13
| { {10, 3, 0}, {6, 6, 1} } | 2 |

14
| { {10, 3, 1} } | 1 |

15
| { {15, 0, 0}, {6, 6, 3} } | 2 |

16
| { {15, 1, 0}, {10, 6, 0}, {10, 3, 3} } | 3 |

A002636 Number of representations of *n* as a sum of up to three nonzero triangular numbers.

- {1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 1, 2, 3, 2, 2, 2, 1, 4, 3, 2, 2, 2, 2, 3, 3, 1, 4, 4, 2, 2, 3, 2, 3, 4, 2, 3, 3, 2, 4, 3, 2, 4, 4, 2, 4, 4, 1, 4, 5, 1, 2, 3, 4, 6, 4, 3, 2, 5, 2, 3, 3, 3, 6, 5, 2, 2, 5, 3, 5, 4, 2, 4, 5, 3, 4, ...}

## Even perfect numbers

Every even perfect number is a **triangular number**, since they are a subset of

where is [necessarily, but not sufficiently] a Mersenne prime.

Every even perfect number is also an hexagonal number, since they are a subset of

where is [necessarily, but not sufficiently] a Mersenne prime.

## See also

- 3-sided polygonal numbers.

- 2-dimensional simplicial polytopic numbers.

## Notes

- ↑ Antonella Cupillari,
*The Nuts and Bolts of Proofs*, Belmont, California: Wadsworth Publishing Company (1989): 13–14.

## External links

- Ken Ono, Sinai Robins, and Patrick T. Wahl, ON THE REPRESENTATION OF INTEGERS AS SUMS OF TRIANGULAR NUMBERS.