### Simplifying non-trivial root expressions

76 days ago

This was a surprising pair of identities I stumbled on in college.

and

So, for example, with a=10 and b=2,

I don’t remember the source of the question, but I suspect it was someone with knowledge of this relation.

I’m very curious whether this is part of a larger class of mathematical expressions involving roots that can be simplified.

Here’s a proof:

Solving for m and n:

Interestingly, this can be used in a telescoping identity when a=k, b=1:

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### Fibonacci-like sequences that converge to 0

1701 days ago

I was curious about whether Fibonacci-like sequences could converge to 0, with a proper choice of starting terms and the usual recurrence relationship:

It turns out that they can, if the ratio of the second term to the first is , or 1-φ, approximately -0.618.

I found this using the closed form for Fibonacci-like sequences

and solving for .

Here are the first few elements of the sequence:

Interestingly, the integer part are the Lucas numbers and the multiples of the square root are the Fibonacci numbers, and their ratio converges to the square root of 5, based on their closed forms.

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### cor: a sample correlation game

1708 days ago

If you know what sample correlation is, you might enjoy Cor: A Sample Correlation Game, which is available here

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### truecrypt on the BeagleBone Black (Debian)

1917 days ago

As I’m using Revision B, I ended up downloading source to a mount point on the sdcard (/mnt/sdcard) as it’s too easy to fill up the 2GB root device otherwise.

export SDCARD=/mnt/sdcard
cd ${SDCARD} wget http://prdownloads.sourceforge.net/wxwindows/wxWidgets-2.8.12.tar.gz tar xvfz wxWidgets-2.8.12.tar.gz export WX_HOME=${SDCARD}/wxWidgets-2.8.12/


mkdir ${SDCARD}/pkcs cd${SDCARD}