Triangular number

Definition

It has certain regularity, arrangements (constituent map), like 1, 3, 6, 10, 15, etc. above these, can represent a triangular shape. Number of total quantities called triangle.

a certain number of points or circle can form an equilateral triangle under the arrangement of the like, such a number is called a triangular number. For example, 10 points can form an equilateral triangle, so 10 is a triangular number:

x

x

x x

x x xx

starting 18 triangular numbers is 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 66, 78, 91, 105, 120, 136, 153, 171 ... (Dany in OEIS A000217)

Nth triangular number is the beginning of N natural number.

All triangular numbers of greater than 3 are not the number.

starting N cube and is the square of the nthly triangular number (for example: 1 + 8 + 27 + 64 = 100 = 10 ² )

The sum of the counts of all triangles is 2.

Any triangular number is multiplied by 8 to add 1 is a square number.

A portion of the triangularities (3, 10, 21, 36, 55, 78 ...) can be represented by the following formula: N × (2n + 1); and the rest of the other part (1, 6, 15, 28, 45, 66 ...) can be represented by N × (2n - 1).

A method for verifying the number of positive integer X is the number of triangularities is calculated:

If n is an integer, then x is the number N number of triangles. If N is not an integer, then x is not a triangular number. This test method is based on Heth 8TN + 1 = S ² N + 1.

Special triangular number

55, 5, 050, 500, 500, 50, 1005,000 ... They are all triangularities.

Number of triangular triangles (66), 1111 Triangle count (617, 716), 111,111 Triangle numbers (6, 172, 882, 716), 11th, 111, 111 triangular numbers (61, 728, 399, 382, ​​716) are all back type triangular numbers However, the 111th, 11th, 111 and 111, 111 triangular numbers are not.

and other number of relationships

The number of four-sided numbers is the number of triangularities in three-dimensional promotion.

The sum of two successive triangularities is the number of squares.

Triangle score is the number of triangular numbers and number of squares.

Triangle count belongs to a polygon number.

All even satisfaction is the number of triangles.

Any natural number is the sum of up to three triangularities. Gauss discovered this law. He wrote in the diary on July 10, 1796: Eyphka! Num = δ + δ + δ

constituent Figure

= 1 s = 1

ooon = 3 s = 6

oooon = 4 s = 10

ooooon = 5 s = 15

... <

According to the natural number of natural numbers, the calculation formula is:

Application

1) Pre-n triangular number of sum: t (n) = s (1) + s (2) + ... + s (n)

by

:

2) Determine if a number is a triangular number: for a positive integer K, if it is a triangle number, there is:

thus:

Specific: Have you noticed, the can carton in the shop window is generally arranged in this way. They are arranged in a triangle in accordance with certain laws. I want to think about: Can you put the nine dots to be placed on a triangle? 9 isn't the triangular shape? Think about it again: Can you put the 25 dots to the above laws into a triangle? 25 Is it a triangular number? In order to easily see the law, we change the triangle number to the figure. Observe these triangles, do you find any laws? The original triangle number is the continuous natural number of natural numbers from L. L is the first triangle number, 3 is the second triangle number, 6 is the third triangle number, 10 is the fourth triangle number, 15 is the fifth triangle number ... So, the seventh triangular number is: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28; the ninth triangle number is: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45; the tenth triangular number is: 1 + 2 + 3 + ... + 10 = 55; the number 100 triangle is: 1 + 2 + 3 + ... + 100 = 5050.

Special case

1.55, 5050, 500500, 50005000 ... are all triangular numbers.

2. Number of triangular triangles (66), 1111 (617716), 111111 (6172882716), 111111 (61728399382716) is back The number of triangles, but 1111, 11111, and 111111 are not.

3. There is still a law in the triangular shape, that is: If the number of all sides is collected in the left to right, you will find that the number interval of each column The same, and all the triangular numbers of the previous column, for example:

15

number of squares

4. For example, 10 points can form an equilateral triangle, so 10 is a triangular number:

The 18 triangular numbers at the beginning of 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253 ...

A triangular number multiply by nine plus one still Is a triangular number.

The number of digits of the triangular number cannot be 2, 4, 7, 9, and the number root cannot be 2, 4, 5, 7, 8.

Triangular square square root is made, which is the number of the triangular number.

Nature

The formula of the Nth triangle number is

Nth triangular number is the N natural number of N natural numbers starting from 1.

All triangular numbers of greater than 3 are not the number.

starting N cube and is the square of the Nth triangular shape (for example: 1 + 8 + 27 + 64 = 100 = 10).

The sum of the counts of all triangles is 2.

Any triangular number is multiplied by 8 to add 1 is a square number.

A portion of the triangle numbers (3, 10, 21, 36, 55, 78 ...) can be represented by the following formula: {\ DisplayStyle N * (2N + 1)}; and the rest Part (1, 6, 15, 28, 45, 66 ...) can be represented by {\ DisplayStyle N * (2N-1)}.

A method for verifying the number of triangularities x is calculated:

If n is an integer, then x is the n triangle number . If n is not an integer, then x is not a triangular number. This test method is based on the constant equity

and other digital relations

• Does at least one of the number of triangular numbers in successive The number below 9000000 is correct.

• The number of four-sided numbers is the number of triangularities in three-dimensional promotion.

• The sum of two successive triangularities is the number of squares.

• The number of triangular squared is the number of triangular numbers and number of squares.

• The number of triangles belongs to a polygon.

• All even the number of dolls is a triangular shape.

• Any natural number is the sum of up to three triangularities. Gauss discovered this rule, and he wrote in the diary on July 10, 1796: Ephka! Num = δ + Δ + δ.