Let S₁=S₂=1, S₃=4, and S_n+3= 2S_n+2+2S_n+1-S_n for n≥1.

Is S_p always a perfect square?

The title suggests the Fibonacci series squared (Leonardo of Pisa, or Leonardo Pisano Fibonacci).

Suppose you have three terms of the Fibonacci series: a, b and a+b. Their squares are a^2, b^2 and (a^2 + b^2 + 2ab).

The next term in the Fibonacci series itself would be a + 2b. Its square is a^2 + 4b^2 + 4ab.

But that square is equal to 2*(a^2+b^2+2ab) + 2b^2 - a^2.  So once we have three consecutive squares of the terms of the Fibonacci series, the next one following the given formula is the square of the next term of the Fibonacci series.

We are given the first three squares of terms in the Fibonacci series, and so by induction, they are all squares of terms of the Fibonacci series.

Posted by Charlie on 2007-03-14 14:15:31