# Difference between revisions of "Pascal's triangle"

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=== Sum of previous values === | === Sum of previous values === | ||

− | One of the best known features of Pascal's Triangle is derived from the combinatorics identity <math>{n \choose k}+{n \choose k+1} = {n+1 \choose k+1}</math>. Thus, any number in the interior of Pascal's Triangle will be the sum of the two numbers appearing above it. For example, <math>{5 \choose 1}+{5 \choose 2} = 5 + 10 = 15 = {6 \choose 2}</math> | + | One of the best known features of Pascal's Triangle is derived from the combinatorics identity <math>{n \choose k}+{n \choose k+1} = {n+1 \choose k+1}</math>. Thus, any number in the interior of Pascal's Triangle will be the sum of the two numbers appearing above it. For example, <math>{5 \choose 1}+{5 \choose 2} = 5 + 10 = 15 = {6 \choose 2}</math>. This property allows the easy creation of the first few rows of Pascal's Triangle without having to calculate out each binomial expansion. |

=== Fibonacci numbers === | === Fibonacci numbers === | ||

− | The [[Fibonacci sequence|Fibonacci numbers]] appear in Pascal's Triangle along the "shallow diagonals." That is, <math>{n \choose 0}+{n-1 \choose 1}+\cdots+{n-\lfloor\frac{n}{2}\rfloor \choose \lfloor \frac{n}{2} \rfloor} = F(n+1)</math>, where <math>F(n)</math> is the Fibonacci sequence. For example, <math>{6 \choose 0}+{5 \choose 1}+{4 \choose 2}+{3 \choose 3} = 1 + 5 + 6 + 1 = 13 = F(7)</math>. A "shallow diagonal" is plotted in the diagram. | + | The [[Fibonacci sequence|Fibonacci numbers]] appear in Pascal's Triangle along the "shallow diagonals." That is, <math>{n \choose 0}+{n-1 \choose 1}+\cdots+{n-\left\lfloor\frac{n}{2}\right\rfloor \choose \left\lfloor \frac{n}{2}\right \rfloor} = F(n+1)</math>, where <math>F(n)</math> is the Fibonacci sequence. For example, <math>{6 \choose 0}+{5 \choose 1}+{4 \choose 2}+{3 \choose 3} = 1 + 5 + 6 + 1 = 13 = F(7)</math>. A "shallow diagonal" is plotted in the diagram. |

=== Hockey-stick theorem === | === Hockey-stick theorem === | ||

The [[Combinatorial identity#Hockey-Stick Identity|Hockey-stick theorem]] states: | The [[Combinatorial identity#Hockey-Stick Identity|Hockey-stick theorem]] states: | ||

− | <math>{n \choose 0}+{n+1 \choose 1}+\cdots+{n+k \choose k} = {n+k+1 \choose k}</math>. Its name is due to the "hockey-stick" which appears when the numbers are plotted on Pascal's Triangle, as shown in the representation of the theorem | + | <math>{n \choose 0}+{n+1 \choose 1}+\cdots+{n+k \choose k} = {n+k+1 \choose k}</math>. Its name is due to the "hockey-stick" which appears when the numbers are plotted on Pascal's Triangle, as shown in the representation of the theorem below (where <math>n=2</math> and <math>k=3</math>). |

+ | |||

+ | <asy> | ||

+ | int chew(int n,int r){ | ||

+ | int res=1; | ||

+ | for(int i=0;i<r;++i){ | ||

+ | res=quotient(res*(n-i),i+1); | ||

+ | } | ||

+ | return res; | ||

+ | } | ||

+ | for(int n=0;n<9;++n){ | ||

+ | for(int i=0;i<=n;++i){ | ||

+ | if((i==2 && n<6)||(i==3 && n==6)){ | ||

+ | if(n==6){label(string(chew(n,i)),(11+n/2-i,-n),p=red+2.5);} | ||

+ | else{label(string(chew(n,i)),(11+n/2-i,-n),p=blue+2);} | ||

+ | } | ||

+ | else{ | ||

+ | label(string(chew(n,i)),(11+n/2-i,-n)); | ||

+ | } | ||

+ | } | ||

+ | } | ||

+ | </asy> | ||

=== Number Parity === | === Number Parity === | ||

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[[Image:Sierpinski.jpg]] | [[Image:Sierpinski.jpg]] | ||

+ | |||

+ | ==== Generalization ==== | ||

+ | This is a special case of Kummer's Theorem, which states that given a prime p and integers m,n, the highest power of p dividing <math>\binom{m}{n}</math> is the number of carries in adding <math>m-n</math> and n in base p. | ||

=== Patterns and Properties of the Pascal's Triangle === | === Patterns and Properties of the Pascal's Triangle === | ||

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==== Rows ==== | ==== Rows ==== | ||

The zeroth row has a sum of <math> 1=2^0 </math>. The first row has a sum of <math> 2=2^1 </math>. The <math>n^{th} </math> row has a sum of <math> 2^n </math> | The zeroth row has a sum of <math> 1=2^0 </math>. The first row has a sum of <math> 2=2^1 </math>. The <math>n^{th} </math> row has a sum of <math> 2^n </math> | ||

+ | |||

+ | === Diagonals === | ||

+ | |||

+ | The 1st downward diagonal is a row of 1's, the 2nd downward diagonal on each side consists of the [[natural numbers]], the 3rd diagonal the [[triangular numbers]], and the 4th the [[pyramidal numbers]]. | ||

==See Also== | ==See Also== |

## Latest revision as of 18:24, 26 March 2020

**Pascal's triangle** is a triangle which contains the values from the binomial expansion; its various properties play a large role in combinatorics.

## Contents

## Properties

### Binomial coefficients

Pascal's Triangle is defined such that the number in row and column is . For this reason, convention holds that both row numbers and column numbers start with 0. Thus, the apex of the triangle is row 0, and the first number in each row is column 0. As an example, the number in row 4, column 2 is . Pascal's Triangle thus can serve as a "look-up table" for binomial expansion values. Also, many of the characteristics of Pascal's Triangle are derived from combinatorial identities; for example, because , the sum of the values on row of Pascal's Triangle is .

### Sum of previous values

One of the best known features of Pascal's Triangle is derived from the combinatorics identity . Thus, any number in the interior of Pascal's Triangle will be the sum of the two numbers appearing above it. For example, . This property allows the easy creation of the first few rows of Pascal's Triangle without having to calculate out each binomial expansion.

### Fibonacci numbers

The Fibonacci numbers appear in Pascal's Triangle along the "shallow diagonals." That is, , where is the Fibonacci sequence. For example, . A "shallow diagonal" is plotted in the diagram.

### Hockey-stick theorem

The Hockey-stick theorem states: . Its name is due to the "hockey-stick" which appears when the numbers are plotted on Pascal's Triangle, as shown in the representation of the theorem below (where and ).

### Number Parity

Consider writing the row number in base two as . The number in the th column of the th row in Pascal's Triangle is odd if and only if can be expressed as the sum of some . For example, . Thus, the only 4 odd numbers in the 9th row will be in the th, st, th, and th columns. Additionally, marking each of these odd numbers in Pascal's Triangle creates a Sierpinski triangle.

#### Generalization

This is a special case of Kummer's Theorem, which states that given a prime p and integers m,n, the highest power of p dividing is the number of carries in adding and n in base p.

### Patterns and Properties of the Pascal's Triangle

#### Rows

The zeroth row has a sum of . The first row has a sum of . The row has a sum of

### Diagonals

The 1st downward diagonal is a row of 1's, the 2nd downward diagonal on each side consists of the natural numbers, the 3rd diagonal the triangular numbers, and the 4th the pyramidal numbers.