Via the Giant’s Shoulders #16, I found Arcsecond: The Renaissance Man Uniform Gravitational Acceleration SMACKDOWN
The post is interestig enough, but what really got me was the following pictoral representation of perfect squares:
If you keep adding up the odd numbers, you get the next perfect square (i.e. sum the quantity (2i-1, from 1 to k, and you get k^2). You see this by adding a new “L” of dots to the previous square, which always has 2 more dots that the previous one, i.e. it’s the next odd number in the sequence, and it makes a new square.
That is so cool! If I had previously known this, I had forgotten it. And I can easily imagine this being taught to me ages ago, and not making quite the same impression because I couldn’t fully appreciate the elegance of it.
Which is why adding up a sequence of the odd numbers from 1 to K will always give you a perfect square. And for each odd number, there is a Pythagorean triplet (a^2+b^2=c^2, where a,b,c are positive integers). 3, 4, 5; 5, 12, 13; 7, 24, 25; 9, 40, 41, and so on.
I wonder if the generalization of that summation for perfect cubes and above has a similarly elegant form involving successive odd numbers
The sum I came up with doesn’t look particularly elegant.. not _as_ elegant, at least.. when it’s less late I might look for something better