Triangular Number

A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n. The sequence of triangular numbers (sequence A000217 in OEIS), starting at the 0th triangular number, is:

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ....

The triangle numbers are given by the following explicit formulas:

T_n= \sum_{k=1}^n k = 1+2+3+ \dotsb +n = \frac{n(n+1)}{2} = {n+1 \choose 2}

where is 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 triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room full of n + 1 total people shakes hands once with each other person. In other words, the solution to the handshake problem of n people is Tn-1.

Triangle numbers are the additive analog of the factorials, which are the products of integers from 1 to n.

The number of line segments between closest pairs of dots in the triangle can be represented with the following recurrence relation:

L_n = L_{n-1} + 3(n-1)

In the limit, the ratio between the two numbers, dots and line segments is

\lim_{n\to\infty} \frac{T_n}{L_n} = \frac{1}{3}

Read more about Triangular Number:  Relations To Other Figurate Numbers, Other Properties, Triangular Roots and Tests For Triangular Numbers

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