Patterns in Pascal's Triangle

Today we got all the way to row 14. The very first row, the point of the triangle that only has one number, is row zero.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 300 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 186 715 1287 1716 1716 1287 715 186 78 13 1

1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1

We expanded on some of the patterns from yesterday and found new patterns. I was very impressed with the pattern-finding. Here are some of the patterns.

The numbers in the second diagonal go up by 1 each time:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

The numbers in the next diagonal go up by 2, then 3, then 4, then 5, etc.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

The numbers in the next diagonal go up by 3, then 6, then 10, then 15, etc.:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

The numbers in row 7, except for the outer 1's, are all multiples of 7. I added that this is true for any prime number row: all the numbers in that row, except the outer 1's, will be multiples of the prime.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 300 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 186 715 1287 1716 1716 1287 715 186 78 13 1

1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1

Row 0 has 1 number, the next row has 2 numbers, the next row has 3 numbers, etc.:

1 - 1 number

1 1 - 2 numbers

1 2 1 - 3 numbers

1 3 3 1 - 4 numbers

Only even-number rows have a middle number (this is because we can split an even number in half, but we can't split an odd number in half).

In even-number rows the "choose 2" number is a multiple of half the row number. In row 4 the number 4C2=6 is a multiple of 2, in row 6 the number 6C2=15 is a multiple of 3, in row 8 the number 8C2=28 is a multiple of 4, etc:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 300 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 186 715 1287 1716 1716 1287 715 186 78 13 1

1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1

The other numbers in that diagonal, on the odd-number rows, are special for a different reason:

3*1=3

5*2=10

7*3=21

9*4=36

etc.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 300 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 186 715 1287 1716 1716 1287 715 186 78 13 1

1 14 91 264 901 2002 3003 3432 3003 2002 901 264 91 14 1