Fibonacci meets Pythagoras

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

F_n^2 + F_{n+1}^2 = F_{2n+1}

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.

7 thoughts on “Fibonacci meets Pythagoras

  1. Nah it’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.

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

  3. Interesting and bizarre.
    Is there an explanation or proof of this phenomenon somewhere?

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