Pascals Triangle Patterns and more Patterns It all

1 And continues with the counting numbers (the Naturals) 1 1 1 10 1

1 … and the Triangular Numbers 1 2 1 1 1 1 10 1

Rows add to powers of 2 1 1 10 1 11 1 13 1

1 Numbers Divisible by 3 1 2 1 1 1 10 1 11 1

1 Numbers Divisible by 5 1 2 1 1 1 10 1 11 1

1 Numbers Divisible by 6 1 2 1 1 1 10 1 11 1

1 Numbers Divisible by 7 1 2 1 1 1 10 1 11 1

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Pascal’s Triangle Patterns, and more Patterns!

It all starts with 1 1 1 2 1 1 1 1 11 1 13 1 14 1 1 15 1001 364 91 455 1365 792 2002 3003 5005 55 715 1 11 66 286 1001 3003 1 10 220 2002 5005 9 165 1287 1 45 495 3003 6435 36 330 1716 1 8 120 792 3432 6435 28 210 924 1 7 84 462 1716 6 56 252 1 21 126 462 1287 715 286 78 495 220 15 70 210 1 5 35 126 330 165 55 10 35 84 4 20 56 120 45 66 12 1 36 9 10 1 21 1 6 15 28 8 3 10 5 7 1 3 4 6 1 12 1 13 78 364 1365 1 91 455 1 14 105 1 15 1

1 And continues with the counting numbers (the Naturals) 1 1 1 10 1 11 12 1 13 1 14 1 1 15 105 220 66 286 455 1365 330 1287 715 2002 3003 1716 5005 6435 55 715 1 11 66 286 1001 3003 1 10 220 2002 5005 9 165 1287 1 45 495 3003 6435 36 330 1716 1 8 120 792 3432 7 84 462 1 28 210 924 3003 21 126 462 1 6 56 252 792 5 35 126 1 15 70 210 495 1001 364 91 165 10 35 84 1 4 20 56 120 45 55 78 28 36 9 6 15 21 1 3 10 5 7 8 3 4 6 1 1 2 1 12 78 364 1365 1 1 13 91 455 1 14 105 1 15 1

1 … and the Triangular Numbers 1 2 1 1 1 1 10 1 11 1 12 1 13 1 14 1 1 15 105 220 66 286 455 1365 330 1287 715 2002 3003 1716 5005 6435 55 715 1 11 66 286 1001 3003 1 10 220 2002 5005 9 165 1287 1 45 495 3003 6435 36 330 1716 1 8 120 792 3432 7 84 462 1 28 210 924 3003 21 126 462 1 6 56 252 792 5 35 126 1 15 70 210 495 1001 364 91 165 55 78 120 45 10 35 84 1 4 20 56 28 36 9 6 15 21 1 3 10 5 7 8 3 4 6 1 12 78 364 1365 1 1 13 91 455 1 14 105 1 15 1

Rows add to powers of 2 1 1 10 1 11 1 13 1 14 1 1 15 105 455 1365 792 2002 3003 5005 55 715 1 11 66 286 1001 3003 1 10 220 2002 5005 9 165 1287 1 45 495 3003 6435 36 330 1716 1 8 120 792 3432 6435 28 210 924 1 7 84 462 1716 6 56 252 1 21 126 462 1287 715 1001 364 330 5 35 126 1 15 70 210 495 220 286 78 91 165 55 10 35 84 1 4 20 56 120 45 66 12 1 28 36 9 6 15 21 1 3 10 5 7 8 3 4 6 1 12 78 364 1365 1 1 13 91 455 1 14 105 1 1= 20 2= 21 4= 22 8= 23 16= 24 32= 25 64= 26 128= 27

1 Even Numbers 1 2 1 1 1 1 10 1 11 1 12 1 13 1 14 1 1 15 105 220 66 286 455 1365 330 1287 715 2002 3003 1716 5005 6435 55 715 1 11 66 286 1001 3003 1 10 220 2002 5005 9 165 1287 1 45 495 3003 6435 36 330 1716 1 8 120 792 3432 7 84 462 1 28 210 924 3003 21 126 462 1 6 56 252 792 5 35 126 1 15 70 210 495 1001 364 91 165 10 35 84 1 4 20 56 120 45 55 78 28 36 9 6 15 21 1 3 10 5 7 8 3 4 6 1 12 78 364 1365 1 1 13 91 455 1 14 105 1 15 1

1 Numbers Divisible by 3 1 2 1 1 1 10 1 11 1 12 1 13 1 14 1 1 15 105 220 66 286 455 1365 330 1287 715 2002 3003 1716 5005 6435 55 715 1 11 66 286 1001 3003 1 10 220 2002 5005 9 165 1287 1 45 495 3003 6435 36 330 1716 1 8 120 792 3432 7 84 462 1 28 210 924 3003 21 126 462 1 6 56 252 792 5 35 126 1 15 70 210 495 1001 364 91 165 10 35 84 1 4 20 56 120 45 55 78 28 36 9 6 15 21 1 3 10 5 7 8 3 4 6 1 1 12 78 364 1365 1 1 13 91 455 1 14 105 1 15 1

1 Numbers Divisible by 5 1 2 1 1 1 10 1 11 1 12 1 13 1 14 1 1 15 105 220 66 286 455 1365 330 1287 715 2002 3003 1716 5005 6435 55 715 1 11 66 286 1001 3003 1 10 220 2002 5005 9 165 1287 1 45 495 3003 6435 36 330 1716 1 8 120 792 3432 7 84 462 1 28 210 924 3003 21 126 462 1 6 56 252 792 5 35 126 1 15 70 210 495 1001 364 91 165 10 35 84 1 4 20 56 120 45 55 78 28 36 9 6 15 21 1 31 10 5 7 8 3 4 6 1 1 12 78 364 1365 1 1 13 14 91 455 1 105 1 15 1

1 Numbers Divisible by 6 1 2 1 1 1 10 1 11 1 12 1 13 1 14 1 1 15 105 220 66 286 455 1365 330 1287 715 2002 3003 1716 5005 6435 55 715 1 11 66 286 1001 3003 1 10 220 2002 5005 9 165 1287 1 45 495 3003 6435 36 330 1716 1 8 120 792 3432 7 84 462 1 28 210 924 3003 21 126 462 1 6 56 252 792 5 35 126 1 15 70 210 495 1001 364 91 165 10 35 84 1 4 20 56 120 45 55 78 28 36 9 6 15 21 1 31 10 5 7 8 3 4 6 1 1 12 78 364 1365 1 1 13 14 91 455 1 105 1 15 1

1 Numbers Divisible by 7 1 2 1 1 1 10 1 11 1 12 1 13 1 14 1 1 15 105 220 66 286 455 1365 330 1287 715 2002 3003 1716 5005 6435 55 715 1 11 66 286 1001 3003 1 10 220 2002 5005 9 165 1287 1 45 495 3003 6435 36 330 1716 1 8 120 792 3432 7 84 462 1 28 210 924 3003 21 126 462 1 6 56 252 792 5 35 126 1 15 70 210 495 1001 364 91 165 10 35 84 1 4 20 56 120 45 55 78 28 36 9 6 15 21 1 31 10 5 7 8 3 4 6 1 1 12 78 364 1365 1 1 13 91 455 1 14 105 1 15 1

And so it goes, on, and on …