**21 Is 35 Of What Number** – This is the second in a series of guest posts by David Benjamin exploring the mysteries of Pascal’s triangle.

Many series of numbers can be found in Pascal’s triangle. The natural numbers are on the second diagonal, which runs in any direction, and the next two diagonals have another important sequence:

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## 21 Is 35 Of What Number

The set of natural numbers is on the second diagonal, the set of triangular numbers is on the third diagonal, and the set of quadratic numbers is on the fourth diagonal.

### Tracing Number 21

, all of which can be represented geometrically by a series of points. Polygon numbers all start with $1$ – representing degenerate polygons that are not really polygons.

Two congruent right triangles can be joined to form an equilateral triangle. Similarly, the sum of consecutive pairs of triangular numbers forms the set of square numbers, and so the sequence of square numbers can also be considered contained in the third diagonal.

Yang Hui invented a formula for finding the sum of triangular numbers. The $nth$ term for the sequence is $frac$ and the sum of the first $n$ terms is $frac$ , where each term or sum is found by permuting $n = 1, 2, 3, 4, . .$

Square numbers can also be found in triangles by calculating the product of six surrounding numbers

### Large Skip Counting Display

Wolfram Mathworld describes the pentatope as the simplest regular figure in four dimensions that represents the four-dimensional analog of the cubic tetrahedron. It is also called a 5-cell because it has five vertices or a pentacoron. A pentagram is a four-dimensional simple.

In Part 1, we saw that the numbers in each row add up to powers of $2$. If we combine the numbers of the first five rows of the triangle, we get a power of $11$.

$1 8 28 56 70 56 28 8 1 rightarrow 1 (8 + 2) (8 + 5) (6 + 7) (0 + 5) (6 + 2) 8 8 1$

Working from the right, add parentheses and take any $10$ as $1$ in the sum of the next parentheses.

#### Sequencing Fibonacci Numbers

When the second number in each row is a prime number, the symmetry of the triangle means that the last number in that row is the same prime number and all the numbers between the two prime numbers are multiples of that prime number.

$55 = 5 times 11$, $165 = 15 times 11$, $330 = 30 times 11$, $462 = 42 times 11$

Start at any $1$ and move diagonally down any rows. The sum of those numbers equals the number on the next row of the opposite diagonal

In the next section, we’ll consider another famous sequence that has an unexpected connection to Pascal’s triangle—the Fibonacci numbers. A patatop number is the number in the fifth cell of any row of Pascal’s triangle starting with the 5-period row 1 4 6. 4 1, either left to right or right to left.

## Find The Lcm Of 21, 35 And 42.

A ptatope of side lgth 5 contains 70 3-spheres. Each level represents one of the first five tetrahedral numbers. For example, the lower (gray) layer has a total of 35 spheres.

Ptatope numbers belong to a class of pictorial numbers that can be represented as regular, discrete geometric patterns.

The formula for the nth ptatope number is the 4th increasing factor of n divided by the factors of 4:

It is a discrete quadrant that can be chosen from among n + 3 objects and is read aloud as “n plus three chooses four”.

#### X×21/4=35/2 Linear Equation

Two out of every three even numbers are also base numbers. To be precise, the (3k – 2)th ptatope number is always the (3k2 – k / 2 )th ptatope number and the (3k – 1)th ptatope number is always the (3k2 + k / 2 )th ptatope number. The (3k)th ptatope number is the Geld ptatope number, taking the negative exponent −3k2 + k / 2 in the formula for pentagonal numbers. (These expressions always return integers).

A prime number is not a predecessor of a prime number (it only needs to check -1 and 4=2

If and only if x is prime and n is a natural number. where x is the nth ptatope number.

In biochemistry, pentatope numbers indicate the number of possible arrangements of the different polypeptide subunits of a tetrameric (tetrahedral) protein.

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