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The natural numbers are also known as the counting numbers.They appear in the second diagonal of Pascal's triangle:







1 








Row 0  






1 

1 







Row 1  





1 

2 

1 






Row 2  




1 
3 
3 

1 





Row 3  



1 

4 

6 

4 

1 




Row 4  


1 

5 

10 

10 

5 

1 



Row 5  

1 

6 

15 

20 

15 

6 

1 


Row 6  
1 

7 

21 

35 

35 

21 

7 

1 

Row 7  
1 

8 

28 

56 

70 

56 

28 

8 

1 
Row 8 
Notice that the numbers in the rows of Pascal's triangle read the same lefttoright as righttoleft, so that the counting numbers appear in both the second left and the second right diagonal.
References
 Ask Dr. Math Archive: Do Natural Numbers Start with 0 or 1?
 Eric's Treasure Trove of Mathematics: Natural Numbers, Counting Numbers.
 Patterns with Pascal  Stacy Boland
 The Twelve Days of Christmas and Pascal's Triangle  Judy Brown
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