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Patterns in Pascal's Triangle - with a Twist - The Solid Downward-Pointing Triangles

Author(s): 
Kathleen M. Shannon and Michael J. Bardzell

 

Here are the triangles for n = 2 through 12 drawn to more rows:

 

Pascal's triangle mod 2 with 125 rows Pascal's triangle mod 3 with 125 rows Pascal's triangle mod 4 with 125 rows
Pascal's triangle mod 5 with 125 rows Pascal's triangle mod 2 with 125 rows Pascal's triangle mod 3 with 125 rows
Pascal's triangle mod 4 with 125 rows Pascal's triangle mod 5 with 125 rows Pascal's triangle mod 2 with 125 rows
Pascal's triangle mod 3 with 125 rows Pascal's triangle mod 4 with 125 rows colors assigned to 0 through 12

You probably noticed immediately the downward-pointing royal blue triangles in all of the patterns. Since royal blue corresponds to the number zero, you can see that, whenever you get a string of zeros on a row, this will generate a downward pointing blue triangle -- any entry below two zeros will be zero. But the non-zero entries on either end of the string will encroach one position on each side for each row.

For example, if you start with a row {1,0,0,0,0,0,1} and generate the following rows, you get:

 

1   0   0   0   0   0   1
  1   0   0   0   0   1  
    1   0   0   0   1    
      1   0   0   1      
        1   0   1        
          1   1          
            2            

The downward pointing triangles containing subsets of the colors for a triangle occur for similar reasons. For example, look at one of the blue triangles for the mod 2 triangle. Now look at the same area in the mod 4 and mod 8 triangles. What do you see? Look at the same region in the mod 6 triangle. What can you say about the colors in this region in the mod n triangles for any even n? (Answer)

Kathleen M. Shannon and Michael J. Bardzell, "Patterns in Pascal's Triangle - with a Twist - The Solid Downward-Pointing Triangles," Loci (December 2004)

JOMA

Journal of Online Mathematics and its Applications

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