<s>[INST] New Golden Ratio (phi) Sequences This is the golden ratio (phi):
$\phi = \sqrt{5}/2 - 1/2 $
If we perform alternate subtraction and addition to ascending powers of phi and its reciprocal we get the sequence of numbers known as the 'Lucas numbers'. 
$1 = 1/\phi - \phi $
$3 = 1/\phi^2 + \phi^2 $
$4 = 1/\phi^3 - \phi^3 $
$7= 1/\phi^4 + \phi^4 $
$11 = 1/\phi^5 - \phi^5 $
...
If we perform alternate addition and subtraction to ascending powers of phi and its reciprocal we get this strange sequence of numbers.
$ \sqrt{5} = 1/\phi + \phi  $
$ \sqrt{5} = 1/\phi^2 - \phi^2 $
$ (2)\sqrt{5} = 1/\phi^3 + \phi^3 $
$ (3)\sqrt{5} = 1/\phi^4 - \phi^4 $
$ (5)\sqrt{5} = 1/\phi^5 + \phi^5 $
...
As you can see in the following examples we can calculate the square or square-root of phi to any power using simple arithmetic. 
If we set the value of x to equal phi, we get the following sequence:
$ x^2 = 1 - x  
\\ x^3 = x - x^2  
\\ x^4 = x^2 - x^3  
\\ x^5 = x^3 - x^4  
\\ x^6 = x^4 - x^5  
\\ x^7 = x^5 - x^6  
\\ x^8 = x^6 - x^7  $
...
$ x^2 = 3/2 - \sqrt{5}/2  
\\ x^3 = 4/2 - (2) \sqrt{5}/2  
\\ x^4 = 7/2 - (3) \sqrt{5}/2  
\\ x^5 = 11/2 - (5) \sqrt{5}/2  
\\ x^6 = 18/2 - (8) \sqrt{5}/2   
\\ x^7 = 29/2 - (13) \sqrt{5}/2  
\\ x^8 = 47/2 - (21) \sqrt{5}/2  $
...
If we set the value of x to equal the reciprocal of phi, we get this second sequence:
$ x^2 = 1 + x  
\\ x^3 = x + x^2  
\\ x^4 = x^2 + x^3  
\\ x^5 = x^3 + x^4  
\\ x^6 = x^4 + x^5  
\\ x^7 = x^5 + x^6  
\\ x^8 = x^6 + x^7  $
...
$ x^2 = 3/2 + \sqrt{5}/2  
\\ x^3 = 4/2 + (2) \sqrt{5}/2  
\\ x^4 = 7/2 + (3) \sqrt{5}/2  
\\ x^5 = 11/2 + (5) \sqrt{5}/2  
\\ x^6 = 18/2 + (8) \sqrt{5}/2   
\\ x^7 = 29/2 + (13) \sqrt{5}/2  
\\ x^8 = 47/2 + (21) \sqrt{5}/2  $
...
Notice in both cases the similarity to the Fibonacci sequence.
Do theses sequence have a name?
 [/INST] Observe these two closed formulas for the Lucas and  Fibonacci numbers, known as Binet's formulas:
$$L_n = \phi^{n} + (-\phi)^{-n}$$
$$F_n = \frac{1}{\sqrt{5}}\left(\phi^{n} - (-\phi)^{-n}\right)$$
Now your observation is:
$$\phi^n = \frac{1}{2}(L_n + F_n\sqrt{5})$$
Plug in the formulas above to verify its correctness. And this is a well-known relationship between the Lucas and Fibonacci numbers and the golden ratio.
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