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TRIANGULAR NUMBER

A triangular number is a number that can be arranged in the shape of an equilateral triangle. The sequence of triangular numbers (sequence A000217 in OEIS) for n = 1, 2, 3... is:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
1 Image:Triangular number 1.png
3 Image:Triangular number 3.png
6 Image:Triangular number 6.png
10 Image:Triangular number 10.png
15 Image:Triangular number 15.png
21 Image:Triangular number 21.png

Since each row is one unit longer than the previous row it can be seen that a triangular number is the sum of consecutive integers.

The formula for the nth triangular number is ½n(n + 1) or (1 + 2 + 3 + ... + [n − 2] + [n − 1] + n).

It is the binomial coefficient

{n+1 \choose 2}

It can also be shown that for any n-dimensional simplex with sides of length x, the formula

\frac {(x)(x+1)\cdots(x+(n-1))} {n!}

yields the number of points that make up the simplex. For example, a tetrahedron with sides of length 2 corresponds to the number (2)(2 + 1)(2 + 2)/6, or 4. The four points forming this configuration are the vertices of the tetrahedron. (Note: A tetrahedron can be created by taking a number, getting the triangle of that number, and then adding to it all the triangles of the numbers before it, so a tetrahedron of 2 would have 2 triangles = 3 plus 1 triangles = 1 = 4.)

One of the most famous triangular numbers is 666, also known as the Number of the Beast. Every perfect number is triangular.

The sum of two consecutive triangular numbers is a square number. This can be shown mathematically thus: the sum of the nth and (n-1)th triangular numbers is {½n(n + 1)} + {½(n − 1)n}. This simplifies to (½n2 + ½n) + (½n2 − ½n), and thus to n2. Alternatively, it can be demonstrated diagrammatically, thus:

16 Image:Square triangle sum 16.png
25 Image:Square triangle sum 25.png

In each of the above examples, a square is formed from two interlocking triangles.

More generally, the difference between the nth m-gonal number and the nth (m+1)-gonal number is the (n-1)th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15.

Also, the square of a triangular number n is the same as the sum of the cubes of the integers 1 to n.

In base 10, the digital root of a triangular number is always 1, 3, 6 or 9. Hence every triangular number is either divisible by three or has a remainder of 1 when divided by nine:

6 = 3×2,
10 = 9×1+1,
15 = 3×5,
21 = 3×7,
28 = 9×3+1,
...

Triangular numbers have all sorts of relations to other figurate numbers. Whenever a triangular number is divisible by 3, one third of it will be a pentagonal number. Every other triangular number is a hexagonal number.

Knowing the triangular numbers, one can reckon any centered polygonal number. The nth centered k-gonal number is obtained by the formula

Ckn = kTn − 1 + 1

where T is a triangular number.

There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36. Some of them can be generated by a simple recursive formula:

S_{n+1} = 4S_n \left( 8S_n + 1\right) with S1 = 1

All square triangular numbers are found from the recursion

Sn = 34Sn − 1 − Sn − 2 + 2 with S0 = 0 and S1 = 1

Two other interesting formulas regarding triangular numbers are:

Ta + b = Ta + Tb + ab

and

Tab = TaTb + Ta − 1Tb − 1,

both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra.

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