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I'm hoping to get some feedback about the two methods I've found for generating the sequences of numbers in the diagonals of Pascal's Triangle.
The first two diagonals can be generated using this method, but they're both pretty trivial and need no algorithm to be generated, so I'll begin with the third diagonal, which is composed of the triangular numbers:
1×3/1=3
3×4/2=6
6×5/3=10
10×6/4=15
15×7/5=21
21×8/6=28
28×9/7=36
36×10/8=45
45×11/9=55
55×12/10=66
Etc.
Next, the fourth diagonal, composed of the tetrahedral numbers:
1×4/1=4
4×5/2=10
10×6/3=20
20×7/4=35
35×8/5=56
56×9/6=84
84×10/7=120
120×11/8=165
165×12/9=220
220×13/10=286
Etc.
Now for the 5-simplex numbers in the 5th diagonal:
1×5/1=5
5×6/2==15
15×7/3=35
35×8/4=70
70×9/5=126
126×10/6=210
210×11/7=330
330×12/8=495
495×13/9=715
715×14/10=1001
Etc.
I'll just do one more because I expect that by now you can see the basic idea
Here are the 6-simplex numbers:
1×6/1=6
6×7/2=21
21×8/3=56
56×9/4=126
126×10/5=252
252×11/6=462
462×12/7=792
792×13/8=1287
1287×14/9=2002
2002×15/10=3003
Etc.
As you can see this is very simple and easy, which is why I don't understand why I have been completely unable to find anything like it at either Google or YouTube. If it has been done before, then great, but if not I don't want it to disappear if my poor health causes me to die suddenly. I'm hoping that someone here can tell me if it has been done, and if so, by whom.
I came up with that about a year ago, but just last week I found a completely different method of doing the exact same thing, this one uses combinations. This one is even simpler and easier than the other, especially since I have this great TI-30XS MultiView, which has a "table" mode, which allows me to input an expression with one variable (x), and returns a list of "y" values. This time I'll go ahead and begin with the first diagonal:
x nCr 0 returns a potentially infinite sequence of 1s.
x nCr 1 returns a list of the positive integers.
x nCr 2 returns a list of the triangular numbers.
x nCr 3 returns a list of the tetrahedral numbers.
x nCr 4 returns a list of the 5-simplex numbers.
x nCr 5 returns a list of the 6-simplex numbers.
And so on for all of the other diagonals.
I don't quite understand why these methods work, nor do I understand why I can find nothing online about them. I'm hoping that someone will give me a clue to both of these mysteries.
The first two diagonals can be generated using this method, but they're both pretty trivial and need no algorithm to be generated, so I'll begin with the third diagonal, which is composed of the triangular numbers:
1×3/1=3
3×4/2=6
6×5/3=10
10×6/4=15
15×7/5=21
21×8/6=28
28×9/7=36
36×10/8=45
45×11/9=55
55×12/10=66
Etc.
Next, the fourth diagonal, composed of the tetrahedral numbers:
1×4/1=4
4×5/2=10
10×6/3=20
20×7/4=35
35×8/5=56
56×9/6=84
84×10/7=120
120×11/8=165
165×12/9=220
220×13/10=286
Etc.
Now for the 5-simplex numbers in the 5th diagonal:
1×5/1=5
5×6/2==15
15×7/3=35
35×8/4=70
70×9/5=126
126×10/6=210
210×11/7=330
330×12/8=495
495×13/9=715
715×14/10=1001
Etc.
I'll just do one more because I expect that by now you can see the basic idea
Here are the 6-simplex numbers:
1×6/1=6
6×7/2=21
21×8/3=56
56×9/4=126
126×10/5=252
252×11/6=462
462×12/7=792
792×13/8=1287
1287×14/9=2002
2002×15/10=3003
Etc.
As you can see this is very simple and easy, which is why I don't understand why I have been completely unable to find anything like it at either Google or YouTube. If it has been done before, then great, but if not I don't want it to disappear if my poor health causes me to die suddenly. I'm hoping that someone here can tell me if it has been done, and if so, by whom.
I came up with that about a year ago, but just last week I found a completely different method of doing the exact same thing, this one uses combinations. This one is even simpler and easier than the other, especially since I have this great TI-30XS MultiView, which has a "table" mode, which allows me to input an expression with one variable (x), and returns a list of "y" values. This time I'll go ahead and begin with the first diagonal:
x nCr 0 returns a potentially infinite sequence of 1s.
x nCr 1 returns a list of the positive integers.
x nCr 2 returns a list of the triangular numbers.
x nCr 3 returns a list of the tetrahedral numbers.
x nCr 4 returns a list of the 5-simplex numbers.
x nCr 5 returns a list of the 6-simplex numbers.
And so on for all of the other diagonals.
I don't quite understand why these methods work, nor do I understand why I can find nothing online about them. I'm hoping that someone will give me a clue to both of these mysteries.
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