The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number. Specifically we have

and so we have the following right triangle.

The hypotenuse will always be irrational because the only Fibonacci numbers that are squares are 1 and 144, and 144 is the 12th Fibonacci number.

It looks like the only Pythagorean triple of this sort is (0,1;1).

https://math.stackexchange.com/questions/1012999/square-fibonacci-numbers

What does F2n+1 the hypotenuese notation mean?

Nah itâ€™s observed by all combinations.

0^2 + 1^2 = 1

1^2 + 1^2 = 2

1^2 + 2^2 = 5

2^2 + 3^2 = 13

…

You always get the odd-numbered numbers in the Fibonacci sequence.

The subscripts are the index of the Fibonacci numbers, so F_n is the nth Fibonacci number.

Interesting and bizarre.

Is there an explanation or proof of this phenomenon somewhere?

Here is a long but explicit proof.

Here’s a sketch of a more elegant proof via Dmitry Rubanovich: “F_{n+m}=F_n F_{m-1} + F_{n+1} F_m. Now set m=n+1 and you get the proof.” See the discussion here.

Interesting