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I have these kinds of expressions:

  • Correct expressions (true and valid) $$\begin{align} 3 &= 3\hbox{ (it's obvious, isn't it?)}\\ 4 &= 1 + 3\\ 8 &= 5\\ 9 &= 1 + 5\\ 12 &= 1 + 3 + 5\\ 18 &= 4 + 6 \end{align}$$

  • True but not valid expressions $$\begin{align} 3 &= 1 + 2\\ 10 &= 4 + 4\\ 12 &= 5 + 3 + 1\\ 14 &= 1 + 4 + 5 \end{align}$$

  • Incorrect expressions (false and invalid) $$\begin{align} 10 &= 4 + 6\\ 11 &= 5 + 6 \end{align}$$

For each non-negative integer in the left side, there is a unique expression on the right side.

Can you tell what is the general rule? I think it is an easy one for you, guys, but feel free to ask for specific values if you need them.

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  • $\begingroup$ What is the difference between true and valid expressions? How can an expression be true, yet not valid? $\endgroup$ – McMagister Aug 12 '15 at 11:04
  • $\begingroup$ A valid expression is written following an specific rule, and a true expression means that both sides have the same value $\endgroup$ – BianB BB Aug 12 '15 at 11:08
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These are the Fibonacci representations of the integers. Valid expressions contain no consecutive or repeated numbers, and are in increasing order.

$3=3=F_3\\ 4=1+3=F_1+F_3\\ 8=8=F_5\\ 9=1+8=F_1+F_5\\ 12=1+3+8=F_1+F_3+F_5\\ 18=5+13=F_4+F_6$

$10=5+5=F_4+F_4$, but it should be written as $2+8=F_2+F_5$ instead. $14=1+5+8=F_1+F_4+F_5$, but it should be written as $1+13=F_1+F_6$ instead.

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  • $\begingroup$ Yes, that's correct $\endgroup$ – BianB BB Aug 12 '15 at 11:40
  • $\begingroup$ Aren't there two 1 at the begining of the Fibonacci $\endgroup$ – Cryol Aug 12 '15 at 11:41
  • $\begingroup$ @Cryol The first one is sometimes counted as $F_0$. $\endgroup$ – f'' Aug 12 '15 at 11:42
  • $\begingroup$ Okay then it makes sense. The Fibonacci was the first Thing i looked at :) $\endgroup$ – Cryol Aug 12 '15 at 11:43
  • $\begingroup$ This is based on the Zeckendorf's theorem $\endgroup$ – BianB BB Aug 12 '15 at 11:45

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