Triangular number

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

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

1
3
6
10
15
21

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

[ ]

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

[ \frac ]

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 even 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 + . This simplifies to (½n² + ½n) + (½n² − ½n), and thus to n². Alternatively, it can be demonstrated diagrammatically, thus:
16
25
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

[Ck_n = kT_+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_ = 4S_n \left( 8S_n + 1\right)] with [S_1 = 1]

All square triangular numbers are found from the recursion

[S_n = 34S_ - S_ + 2] with [S_0 = 0] and [S_1 = 1]

Two other interesting formulas regarding triangular numbers are:

[T_ = T_ + T_ + ab]

and

[T_ = T_T_ + T_T_],

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

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