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Yes. Who says the sort of numbers or sequences we think are so obvious and fundamental are obvious and fundamental to an alien intelligence? (It would be even more interesting if they used the octal base...)

It's just 2sqrt(10)/3... not any fundamental constant

The golden ratio is "just" (sqrt(5)+1)/2, not any fundamental constant. But it's still called golden ratio.

If the sides are A, B, C, what are the sides AB, AC, BC?

I assume that you mean that the angles are A,B,C and the side lengths are opposite the angles?

Isn't it a bit unlikely that aliens would use the decimal system?

I think that would depend on their number of appendages, and the number of digits per appendage. Assuming a trunk/appendage/digit physiology, of course.

Since they'd be sending a countable set of beeps, it wouldn't matter what base they used since binary 1000 would have the same number of beeps as decimal 8

That works if the data is a sequence of distinct numbers. If the sequence is supposed to be the digits of a number, like the decimal expansion of sqrt(3), then the base matters.

Sending natural numbers as a sequence of pulses is easy. Sending real numbers is much harder.

Why not use the continuing fraction denominators? Those are all whole numbers.

By the way, that's one example of how hard it would be to understand an alien communication. What if they have developed a math system using continued fractions and not decimals (or octals, or binary numbers...)? Would we be able to figure it out?

OEIS.org is your friend. And it seems like it's also Mike's friend, since the text matches almost by 100% :) http://oeis.org/A195284 Just try typing random sequences of eight integers and see how often you get a match.