Triangular number

A triangular figure is a figure which corresponds to the sum of all numbers from 1 to an upper limit. For example, the number 10 is a triangle, as is. The first triangular numbers are

For some authors, the zero is not a triangular number, so the number sequence only begins with one.

The term triangular number is derived from the geometric figure of the triangle. The number of stones that you need to insert an equilateral triangle is always equal to a triangular number. From ten stones a triangle for example, can put in which each side of four stones is formed.

Because of this relationship with a geometric figure are the triangular numbers to the figurate numbers, which also include the square numbers and cubic numbers belong. Even Pythagoras has dealt with triangular numbers.

Calculation

The triangle -th number is the sum of the numbers from 1 to.

Instead of adding the individual numbers, triangular numbers can be calculated by the Gaussian sum formula.

This formula is the same as the above second binomial

This formula can be illustrated by designing the triangular number. The triangular number can be interpreted as a triangle or stairs. Twice a triangular number corresponds to two equal steps, which can be taped together to form a rectangle.

This rectangle is high balls and balls wide and thus includes balls. A triangular number is equal to half of the balls, from which the above formula is obtained for triangular numbers.

Properties

• For all triangular numbers > 3 is a composite number.

• The sum of the first n cubes is equal to the square of the nth triangular number [eg: 1 8 27 64 = 100 = 102 ]
• The difference of the squares of two consecutive triangular numbers results in a cubic number. This can be derived from the above continuous property. If the square of the nth triangular number is formed from the sum of the first n cubes, and the square of the ( n 1 ) th triangular number is formed from the sum of the first n 1 cubes, must as a difference, the ( n 1 )-th cube of coming out.

• Eight times a triangular number added to 1 always yields an odd square number:

• Every even perfect number is a triangular number: After Leonhard Euler, a just perfect number can be represented by the formula, where a prime ( Mersenne prime ) must be. If you multiplicatively with 2 extends the formula, and substituted, one arrives at the formula that represents the triangular numbers:

Sum of three triangular numbers

Pierre de Fermat presented on the assumption that every natural number can be represented as a sum of at most three triangular numbers. This conjecture was proved by Carl Friedrich Gauss, who wrote in his diary on July 10, 1796

The more general statement is known as Fermat Polygonalzahlensatz.

Relations with square numbers

Sum of two consecutive triangular numbers

The sum of two consecutive triangular numbers gives a square number. The picture shows an example of how the triangular numbers and square number 25 add.

This phenomenon can also be described by a formula.

For another explanation of this phenomenon breaks the triangular number in the sum of and the previous triangular number. Accordingly applies

The fact that two consecutive triangular numbers add up to a perfect square, was already in the 2nd century by the Greek mathematician Theon of Smyrna in his work " The on mathematical knowledge for reading Plato Useful" written down.

Alternating sum of square numbers

Taking the square number and subtracts and adds alternately the smaller squares, then the result obtained is the -th triangular number. For example, the fourth and fifth triangular number calculated as follows:

By making use of the fact that every square number is the sum of two consecutive triangular numbers, can you explain this connection based on its geometrical illustration.

It is seen that with the exception of the largest each triangle occurs in the sum exactly twice: once each with pluses and minuses. Thus, the small triangles to reduce the sum and is left is the large triangle.

With the help of mathematical vocabulary can be above facts reflect very briefly: the -th triangular number is the alternating sum of the squares of 1 to the corresponding mathematical formula is

Square - triangular numbers

Square - triangular numbers are triangular numbers that are square numbers simultaneously. The first square - triangular numbers are

These are the triangular numbers with the indices

Thus, a triangular number can be a square number, you have for this number

The following shall apply: Of the two expressions, and must be one of the two is an odd square number. while the other must be double a square number. Suppose one of the two expressions, or was just a square number and the other expression twice a square number. This leads to a contradiction, since twice any square number is an even number. An even number plus one, but must be in an odd number, but that did not do any of our consideration. So must be an odd square number of an expression.

Triangular Numbers and centered Polygonalzahlen

Centered Polygonalzahlen associated with regular polygons that are placed according to the following pattern: a single stone is located in the center of the polygon. To this stone more polygons are added, wherein the side lengths increase from the inside outwards in each case by one.

These patterns can also be placed according to a different rule. Re- starting with the individual stone in the middle. But in the second step are wrapped for - te - centered Eckszahl triangles according to the pattern of the -th triangular number around the center. The following figure shows this for the first to fourth centered square number.

It follows for the -th - centered Eckszahl following formula:

Zahlenpalindrome under the triangular numbers

Among the triangular numbers, there are several Zahlenpalindrome. Examples are

Among these, the 11th, the 1.111te that 111.111te and 11.111.111te triangular number. From the 1.111ten and the triangular number 111.111ten Charles Trigg has found that it is Zahlenpalindrome.

Series of the reciprocals

Is the sum of the reciprocal values ​​of all the triangular numbers

Solution by Gottfried Wilhelm Leibniz, with

Triangle root

Analogous to the square root in the square number n can be x determine the side length of a triangular number with the root triangle:

Thus, for example, the Dreieick number x = 10 is formed from n = 4 rows.

Miscellaneous

• The tenth, hundredth, thousandth, ten-thousandth, etc. triangular number is 55, 5,050, 500,500, etc. 50.005 million ( OEIS, A037156 ).

• The triangle strips are divided into two sub-sequences. The terms of the sequence 3, 10, 21, 36, 55, 78, ... ( OEIS, A014105 ) can be calculated using the equation form (see also Sophie Germain prime ). For the other half, the Sechseckszahlen 1, 6, 15, 28, 45, 66, ... ( OEIS, A000384 ), the formation rule.
• The Puerto Rican mathematician Pedro Antonio Pizá was 1950, the relationship

• Is the number of diagonals in the convex -gon.

Generalizations

→ Main article: Polygonalzahl regular figured number

There are essentially two generalizations of triangular numbers. If one stays in the plane, then you can apply the design principle of the triangular numbers to polygons with more vertices. This creates the Polygonalzahlen, which include for example the square numbers and pentagonal count.

The second generalization is to leave the level and move on to higher dimensions. In three dimensions, then one considers a tetrahedron is a pyramid with equilateral triangles as sides. In the Four-dimensional leads to the Pentatop whose sides are tetrahedra. This goes on and on. The associated figurate numbers mean tetrahedral numbers, Pentatopzahlen and in the general case regular figured numbers. In the one-dimensional are still the natural numbers to mention.