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Triangular numbers

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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, ...}

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 of as a sum of three 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 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

Notes

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

External links