Two-thirds of all Fibonacci numbers are…

Today’s New York Times crossword puzzle (October 28, 2020), by Peter Gordon, 34 across. Three letters, “like two-thirds of all Fibonacci numbers”.

The answer is this sequence.

To get the Fibonacci numbers you start with 1 and 1, and then each number is the sum of the two before it. I’ve bolded the even numbers.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

So it’s not just some random smattering of these numbers that happens to be odd, but every third one. To see this, let’s build an addition table for odd and even numbers:

+evenodd
evenevenodd
oddoddeven
Addition table for even and odd numbers

Then if you start with two odd numbers, just following this gets you

odd, odd, even, odd, odd, even, …

and this will repeat itself forever. (If you started with “odd, even” or “even, odd” you’d get the same pattern, but shifted; if you start with “even, even” then the sequence stays even forever.

(The Twitter hashtag #NYTXW mostly missed this, preferring to focus on, and complain about, the fact that the puzzle was built around a too-long quote from Sex and the City.)

Note that 34 (the position of this clue in the puzzle) is a Fibonacci number. I’d like to think this was intentional.

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