Proof

Let $$S = \sum_{r=1}^n r^2$$.

Then $$S = \\ 1+\\ 2+2+\\ 3+3+3+\\ \cdots \\ n+n+n+\cdots+n$$

Also, $$S = \\ n+\\ n+(n-1)+\\ n+(n-1)+(n-2)+\\ \cdots \\ n+(n-1)+(n-2)+\cdots+2+1$$

And $$S = \\ n+\\ (n-1)+n+\\ (n-2)+(n-1)+n+\\ \cdots \\ 1+2+3+\cdots+(n-1)+n$$

So $$S+S+S = \\ (2n+1)+\\ (2n+1)+(2n+1)+\\ (2n+1)+(2n+1)+(2n+1)+\\ \cdots \\ (2n+1)+(2n+1)+(2n+1)+\cdots+(2n+1)+(2n+1)$$

i.e., $$3S = (2n+1) \times \frac{n(n+1)}{2}$$ or $$S = \frac{1}{6} n(n+1)(2n+1)$$ (via Jeremy Kun)

“Why isn’t the new year on winter solstice?”

The answer, honestly, is that the Romans had no fucking idea how to run a calendar.

Having trouble with MastodonToTwitter.py - don't know where to turn for help.

File
"/usr/local/lib/python3.4/dist-packages/mastodon/Mastodon.py", line 963, in __api_request

if url['rel'] == 'next':

KeyError: 'rel'

Help?

A young man defeats several martial arts masters by inventing a martial art that depends solely on his infallibly good luck. He names it Bullshitsu.
#writingprompts #writing

If any tooters want to get an article in issue 06 of Chalkdust, get writing! chalkdustmagazine.com/write-fo

Is $$\sum_{r=100}^{0} r$$:

a) 5050
b) 0
c) -5050
d) other

I have an Opinion, but I don't want to share it until I know whether it's controversial (or worse yet, Wrong.)

@ColinTheMathmo I would consider a delay in email a feature, not a bug. People seem to confuse email with instant messaging. Email is supposed to be asynchronous. I guess some instant messaging tools are also supposed to be asynch, but apparently everything should be like SMS.

No, wait, got it, it's in the column settings.

Is there an easy way to turn off the 'bloop' notifications? I can't see it in the settings.

Let $$C_1$$ and $$C_2$$ be circles that intersect at right angles, having $$P_1$$ and $$P_2$$ as their centres. Then the circle with diameter $$\vec{P_1 P_2}$$ has a quarter of the area of the sum of the areas of $$C_1$$ and $$C_2$$. The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!