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Birthday dates using pi continued fractions

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For π Day. 2015, Wolfram Research stirred up publicity with the blog "Pi or Pie?! Celebrating Pi Day of the Century (And How to Get Your Very Own Piece of Pi)":

https://blog.stephenwolfram.com/2015/03/pi-or-pie-celebrating-pi-day-of-the-centuryand-how-to-get-your-very-own-piece-of-pi

I grumbled that no self-respecting Deity would bother sending clues to worshipers dumb enough to use decimal instead of continued fractions (CF). However, in 2019 we're barely able to afford a full CF version of Wolfram's birthday games. Assuming Gauss–Kuzmin distribution, define

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Then

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I.e, 41.5% of terms should be 1, 17% should be 2, etc. But cfprob also gives us the probabilities of term sequences:

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(Invariant under reversal but not shuffling.)

This says to expect about six 1,2,3's in every burst of 1000 terms:

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Try a million :

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(0.5 seconds for a million terms. Have I actually lived to see this?)

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Continued fractions accommodate fancier date formats:

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Sure enough, there were two of them.

But for really fancy dates,

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we'll need Eric Weisstein's record π CF calculation.

Utility functions

These are the definitions of the utility function tim you will need for the above evaluations:

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