Triangular numbers

This article needs more work.

Please help by expanding it!

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

$t_n:=\sum _{i=1}^{n}i=\frac{n(n+1)}{2}=(\genfrac{}{}{0ex}{}{n+1}{2}),n\ge 0,$

where $t_0=0$ since it is the empty sum of positive integers (giving the additive identity, i.e. 0), and $(\genfrac{}{}{0ex}{}{n}{k})$ is a binomial coefficient. The $n$th triangular number is thus one half of the $n$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 $n$ positive integers (the $n$th triangular number $t_n$) is equal to $\frac{n^2+n}{2}$.

Proof. (Gauß) We can write $t_n=1+2+3+\dots +(n-1)+n$. Since addition is commutative, we can also write $t_n=n+(n-1)+\dots +3+2+1$. If we add up these expressions term by term, left to right, we obtain $2t_n=(1+n)+(2+(n-1))+\dots +((n-1)+2)+(n+1)$. Each of these parenthesized addends works out to $n+1$ and there are $n$ of these addends. Therefore, $2t_n=n(n+1)=n^2+n$ and dividing both sides by 2 we get $t_n=\frac{n^2+n}{2}$ as specified by the theorem. □

Relations

$t_n-t_{n-1}=n,n\ge 1.$

$t_n-2t_{n-1}+t_{n-2}=1,n\ge 2.$

$t_n+t_{n-1}=n^2,n\ge 1.$

$\frac{t_n}{t_{n-1}}=\frac{n+1}{n-1},n\ge 1.$

Relations involving binomial coefficients

Charles Marion <[email protected]> suggested the following two relations:

$\sum _{k=0}^m(-1{)}^k(\genfrac{}{}{0ex}{}{m}{m-k})t_{n-k}=0,n\ge m>2,$

$\sum _{k=0}^m(\genfrac{}{}{0ex}{}{m}{m-k})t_{n-k}=\sum _{k=0}^{m-1}(\genfrac{}{}{0ex}{}{m-1}{m-k-1})(n-k{)}^2,n\ge m>0,$

where $m=1$ gives $t_n+t_{n-1}=n^2.$

Sum of reciprocals

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

$\sum _{i=1}^n\frac{1}{t_i}=\frac{n^2}{t_n}.$

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

$\underset{n\to \mathrm{\infty }}{\lim }\sum _{i=1}^n\frac{1}{t_n}=\underset{n\to \mathrm{\infty }}{\lim }\frac{n^2}{t_n}=\underset{n\to \mathrm{\infty }}{\lim }\frac{2n^2}{n(n+1)}=2.$

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 $n$ as a sum of three triangular numbers

$n$	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

$2^{n-1}\cdot (2^n-1)=\frac{(2^n-1)2^n}{2}=t_{2^n-1},$

where $2^n-1$ is [necessarily, but not sufficiently] a Mersenne prime.

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

$2^{n-1}\cdot (2^n-1)=2^{n-1}\cdot (2\cdot 2^{n-1}-1)=h_{2^{n-1}},$

where $2^n-1$ is [necessarily, but not sufficiently] a Mersenne prime.

See also

• 3-sided polygonal numbers.

• 2-dimensional simplicial polytopic numbers.

External links

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