Now, going back to our original sequence of square numbers 1, 4, 9, 16, 25,..... we notice that the difference between successive terms creates the sequence 3, 5, 7, 9,..... Furthermore, the difference of successive terms for this sequence yields 2, 2, 2, 2, 2,.....i.e,
Running the same algorithm with the sequence of cubes we get the following:
This pattern, if there actually is one, struck me as something that might be very interesting to investigate. So, I ran the same algorithm with powers to the fourth, fifth, sixth, etc.
Let's begin by looking at a spreadsheet with both the first power and the square numbers . See spreadsheet.
You can see that for the first power, the differences converge to 1 on the level of the first difference taken. For the squares, the differences converge to 2, on the second level of differences.
Let's look at a spreadsheet showing the algorithm on cubes. See spreadsheet.
We can see that for the sequence of cubes we get differences that converge to 6 on the third level of differences. I see a pattern forming, but let's go further just to make sure.
Let's look at a spreadsheet showing the fourth powers. See spreadsheet.
We can see from the spreadsheet that the sequence of fourth powers has differences that concerge to 24 on the fourth level of differences.
Seeing this pattern, I would say that for the sequence of fifth powers we will get differences converging to 120 on the fifth level of differences. Let's view a spreadsheet to see this. See spreadsheet.
We can see from the spreadsheet that, indeed, the sequence of fifth powers had differences that converge to 120 on the fifth level of differences.
The pattern is now clear:
To see a complete array of triangles like we saw above click here.
Looking at the triangles, there are many different patterns that one can see, similarly to how Paschal's triangle has several facets. One particular pattern that is interesting to see is the fact that the difference of any two consecutive numbers of power n is always ODD! In fact, within any paricular triangle only the second row contains odd numbers, all subsequent rows contain only even numbers.
I wish that I had a proof of this to show, but I have not come up with one at this time. I am, however, working on a proof of this as we speak.