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Triangular numbers
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 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.