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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?

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    $\begingroup$ You have "discovered" one of the linkages between Fibonacci and Lucas numbers. Still a good observation. $\endgroup$ Jan 28 '18 at 1:26
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    $\begingroup$ Good observation for sure and $+1$ $\endgroup$ Jan 28 '18 at 4:17
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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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    $\begingroup$ Can you please look at my newest post. Thank you. $\endgroup$ Feb 18 '18 at 4:04

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