So, in the previous post, I discussed Pascal's Triangle, and specifically how many digits it takes to write out a given row in the triangle. If DL(n) is the number of digits it takes (not counting spaces and things), it has an explicit formula as
However, that is an extremely problematic function. Floors? Logs? It gets ridiculous. So it becomes necessary to determine useful bounds to approximate the function with. In the previous post, I was able to determine the following lower bound, that
And I just discovered that Mathematica has a LaTeX copy function : ) That is quite handy.
Now, that lower bound was actually determined relatively carefully, while the upper bound I determined basically by making stuff up.
However, using the same approach to the upper bound I used on the lower bound, you work out the following, that
Plotting the two bounds, they look quite pretty.
Looking at that graph, two things are relatively clear. One, the plot of DL(n) in the middle demonstrates that the function would be relatively difficult to calculate explicitly, jumping and bumping about like that. Two though, it's interesting the extent to which DL(n) seems pegged relatively right in the middle of the two bounds. Looking at a plot of DL(n) and the average of the two bounds, it becomes even more interesting.
Just looking at that plot, it's clear that the average of the two bounds is a REALLY good approximation of the function itself. That plot goes out n = 40. Going out to n = 300, it's practically impossible to tell the two functions apart.
And, even better, if we call this function J(n), it has a really nice form!
Given how nasty DL(n) is by itself, J(n) is indeed -quite- nice.
And here's a plot of the error, J(n) - DL(n),
Which is all kinds of interesting. It grows exceptionally slowly. The error for the 300th line is only 60 or so digits, which is quite impressive considering that DL(300) = 19329. Notice too, that it seems exceptionally linear, and also seems to have some kind of embedded periodic activity.
Very very interesting.
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