I love numbers.
Warning: You opened the floodgates. My Aspie-special-interest speech follows.
All my favorites are small composites, as well as 0, 1, and the oddest prime (that is 2 for you non math people). This is because I love seeing how they fit together, and because of this I can become fascinated by series.
The below is an aspect of a set of series that Ive been tinkering with off and on the last several months.
Code:
0^3 1^3 2^3 3^3 4^3 5^3
line 0: 0 1 8 27 64 125
line 1: 1 7 19 37 61
line 2: 6 12 18 24
line 3: 6 6 6
The recursive difference of adjacent terms of the first 6 terms of [img]http://www.bgoncalves.com/cgi-bin/mimetex/mimetex.cgi?\sum%20_{k=1}^{+\infty}%20(k-1)^n[/img] when n=3 results in the series {6,6,6} on line 3 before going to all 0's on successive, defined lines. (line k and after are undefined)
This is actually because because (3!)=6. Try adjusting n to another integer, you'll see that line n is only solely populated by (k-n) terms of (n!).
Note the figurative number series, and the distinctly linear plot of the last two lines. Brings to mind the process of differentiation, but it leaves me with a hint of a thought of an alternative application (differentiation of a series of differences of powers, and polyhedral numbers expressed in the same place?).
Ive been trying to find a correlation between the first (or second) term on each successive line. For n=3, it is {0,1,6,6} and {1,7,12,6} that I am most interested in. I want to be able to know those series for any n.
This works for all positive integer values of n and k. And its weird, because its really just binomial expansion... but so alluring and complex.
Hmm, need to study for the math class I am actually in, but I want to write a program to help me visualize some of these patterns.
Need trumps want.