Why is 18 = 4 + 6? Find the general rule

Question

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.

What is the difference between true and valid expressions? How can an expression be true, yet not valid? — McMagister, Aug 12 '15 at 11:04
A valid expression is written following an specific rule, and a true expression means that both sides have the same value — BianB BB, Aug 12 '15 at 11:08

f'' · Accepted Answer · 2015-08-12 11:41:00Z

9

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.

answered Aug 12 '15 at 11:36

f''

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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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