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## Tuesday, May 4, 2010

A reader ( Paddy ) surprised me with an excellent answer to the exercise I posted. The series I posted was in fact the running total of the squares of the Fibonacci numbers. Let me put that in a table.

$\begin{matrix} n & F(n) & (F(n))^2 & \Sigma\\ 1 & 1 & 1 & 1\\ 2 & 1 & 1 & 2\\ 3 & 2 & 4 & 6\\ 4 & 3 & 9 & 15\\ 5 & 5 & 25 & 40\\ 6 & 8 & 64 & 104\\ 7 & 13 & 169 & 273 \end{matrix}$

$\begin{matrix} n & F(n) & F(n) * F(n+1)\\ 1 & 1 & 1 \\ 2 & 1 & 2 \\ 3 & 2 & 6 \\ 4 & 3 & 15 \\ 5 & 5 & 40 \\ 6 & 8 & 104 \\ 7 & 13 & 273 \end{matrix}$

So $\sum_{k=1}^{n} F_n^{2} = F_n * F_{n+1}$
An interesting conjecture to prove formally.

#### 1 comment:

If we look at Fn * Fn+1
Fn+1 = Fn + Fn-1
So
Fn * (Fn + Fn-1)
= Fn^2 + Fn * Fn-1
Fn = Fn-1 + Fn-2 so we have
Fn^2 + Fn-1 * (Fn-1 + Fn-2)
= Fn^2 + Fn-1^2 + Fn-1 * Fn-2
etc, etc.
Works?

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Mathematics: is it the fabric of MEST?
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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)