User:Continuedfraction

Note
All of these formulae I found are base on experimental just using a basic scientific calculator,

I have no ideas how to proof it.

user:case2009

user:limitphi

Example 1
$$\phi^2=\left(2-\frac{1}{\left(2-\frac{1}{\left(2- \frac{1}{2-\dots}\right)^2}\right)^2}\right)^2=2.61803...$$

This is how I work it out.

$$\left(2-\frac{1}{2}\right)^2=A_1={2.25}$$

$$\left(2-\frac{1}{A_1}\right)^2=A_2\approx{2.419}$$

$$\left(2-\frac{1}{A_2}\right)^2=A_3\approx{2.517}$$

$$\left(2-\frac{1}{A_3}\right)^2=A_4\approx{2.56}$$

$$\left(2-\frac{1}{A_4}\right)^2=A_5\approx{2.59}$$

$$\left(2-\frac{1}{A_5}\right)^2=A_6\approx{2.606}$$

$$\left(2-\frac{1}{A_6}\right)^2=A_7\approx{2.612}$$

$$\left(2-\frac{1}{A_7}\right)^2=A_8\approx{2.615}$$

$$\left(2-\frac{1}{A_8}\right)^2=A_9\approx{2.616}$$

$$\left(2-\frac{1}{A_9}\right)^2=A_10\approx{2.617}$$

Example 2
$$(\phi^4-1)^2=\left(6-\frac{5}{\left(6-\frac{5} {\left(6-\frac{5}{6-\dots}\right)^2}\right)^2}\right)^2=34.270509...$$

This is how I work it out.

$$\left(6-\frac{5}{6}\right)^2=A_1\approx{26.69}$$

$$\left(6-\frac{5}{A_2}\right)^2=A_3\approx{33.78}$$

$$\left(6-\frac{5}{A_3}\right)^2=A_4\approx{34.24}$$

$$\left(6-\frac{5}{A_3}\right)^2=A_5\approx{34.26}$$

$$\left(6-\frac{5}{A_4}\right)^2=A_6\approx{34.27}$$

$$\left(6-\frac{5}{A_5}\right)^2=A_7\approx{34.2705}$$

$$\left(6-\frac{5}{A_6}\right)^2=A_8\approx{34.270509}$$

Example 3
$$2\phi=\sqrt{8+\frac{8}{\sqrt{8+ \frac{8}{\sqrt{8+\frac{8}{8+\cdots}}}}}}=3.23606...$$

This is how I work it out.

$$\sqrt{8+\frac{8}{8}}=A_1={3}$$

$$\sqrt{8+\frac{8}{A_1}}=A_2\approx{3.2659}$$

$$\sqrt{8+\frac{8}{A_2}}=A_3\approx{3.2325}$$

$$\sqrt{8+\frac{8}{A_3}}=A_4\approx{3.2364}$$

$$\sqrt{8+\frac{8}{A_4}}=A_5\approx{3.23601}$$

$$\sqrt{8+\frac{8}{A_5}}=A_6\approx{3.23607}$$

$$\sqrt{8+\frac{8}{A_6}}=A_7\approx{3.236067}$$

Definition
$$\phi=\frac{1+\sqrt{5}}{2}$$

Ln ; Lucase numbers

'''L1 = 2 L2 = 1 L3 = 3 ... L(N+2) = LN + L(N+1) '''

Fn ; Fibonacci numbers

'''F1 = 1 F2 = 1 F3 = 2 ... F(N+2) = FN + F(N+1) '''

Quadratic equation
$$x^2-nx-n=0$$

The solution to this quadratic equation is

$$x^2=\left(\frac{n+\sqrt{n(n+4)}}{2}\right)^2=\left(n+1-\frac{n}{\left(n+1-\frac{n}{\left(n+1-\frac{n}{n+1-\dots}\right)^2}\right)^2}\right)^2$$

$$\phi^2=\left(2-\frac{1}{\left(2-\frac{1}{\left(2-\frac{1}{2-\dots}\right)^2}\right)^2}\right)^2$$

$$(2+2\sqrt{2})^2=\left(5-\frac{4}{\left(5-\frac{4}{\left(5-\frac{4}{5-\dots}\right)^2}\right)^2}\right)^2$$

special case
$$(\phi^4-1)^2=\left(6-\frac{5}{\left(6-\frac{5}{\left(6-\frac{5}{6-\dots}\right)^2}\right)^2}\right)^2$$

$$\frac{-1+\sqrt{21}}{2}=\sqrt{6-\frac{5}{\sqrt{6- \frac{5}{\sqrt{6-\frac{5}{6-\cdots}}}}}}$$

$$5=6-\frac{5}{6-\frac{5}{6-\cdots}}$$

root
$$\sqrt{2+\sqrt{3}}=\sqrt{2}+\frac{1}{\sqrt{2}+\frac{1}{\sqrt{2}+\cdots}}$$

General Formula

$$\sqrt{k+1+\sqrt{2k+1}}=\sqrt{2}+\frac{k}{\sqrt{2}+\frac{k}{\sqrt{2}+\cdots}}$$

Pythagoras triples
$$\sqrt{5+3}=\sqrt{2}+\frac{4}{\sqrt{2}+\frac{4}{\sqrt{2}+\cdots}}$$

$$\sqrt{13+5}=\sqrt{2}+\frac{12}{\sqrt{2}+\frac{12}{\sqrt{2}+\cdots}}$$

$$\sqrt{25+7}=\sqrt{2}+\frac{24}{\sqrt{2}+\frac{24}{\sqrt{2}+\cdots}}$$

$$\sqrt{c+a}=\sqrt{2}+\frac{b}{\sqrt{2}+\frac{b}{\sqrt{2}+\cdots}}$$

Where

$$c^2=a^2+b^2$$

Integer
$$1=\sqrt{2}+\frac{1-\sqrt{2}} {\sqrt{2}+\frac{1-\sqrt{2}}{\sqrt{2}+\frac{1-\sqrt{2}}{\sqrt{2}+...}}}$$

$$k=\sqrt{2}+\frac{k^2-k\sqrt{2}} {\sqrt{2}+\frac{k^2-k\sqrt{2}}{\sqrt{2}+\frac{k^2-k\sqrt{2}}{\sqrt{2}+...}}}$$

$$-k=\sqrt{2}+\frac{k^2+k\sqrt{2}} {\sqrt{2}+\frac{k^2+k\sqrt{2}}{\sqrt{2}+\frac{k^2+k\sqrt{2}}{\sqrt{2}+...}}}$$

General Formula

$$n\sqrt{y}=\sqrt{y}+\frac{yn(n-1)}{\sqrt{y}+ \frac{yn(n-1)}{\sqrt{y}+\frac{yn(n-1)}{\sqrt{y}+\cdots}}}$$

phi

$$\frac{\sqrt{2}}{\phi}=\frac{2} {\sqrt{2}+\frac{2}{\sqrt{2}+\frac{2}{\sqrt{2}+...}}}$$

$$n\phi=n+\frac{n^2}{n+\frac{n^2}{n+\frac{n^2}{n+\cdots}}}= \sqrt{n^2+n\sqrt{n^2+n\sqrt{n^2+n+\cdots}}}$$

$$n\phi^2=n+\frac{\phi^3{n^2}}{n+\frac{\phi^3{n^2}}{n+\frac{\phi^3{}n^2}{n+\cdots}}}= \sqrt{{\phi^3}n^2+n\sqrt{{\phi^3}n^2+n\sqrt{{\phi^3}n^2+n+\cdots}}}$$

$$\frac{n+\sqrt{n(n+4)}}{2}=n+\frac{n}{n+\frac{n}{n+\frac{n}{n+\cdots}}}$$

$$\frac{\sqrt{1}+\sqrt{3}}{\sqrt{2}}=\sqrt{1+\sqrt{2}\sqrt{1+\sqrt{2}\sqrt{1+\cdots}}}$$

$$\frac{1+\sqrt{2k+1}}{\sqrt{2}}=\sqrt{k+\sqrt{2}\sqrt{k+\sqrt{2}\sqrt{k+\cdots}}}$$

$$n^2(n+1)=\frac{[n(n+1)]^3}{n(n+1)+\frac{[n(n+1)]^3}{n(n+1)+\cdots}}$$

General formulae

$$2n^2=\frac{[n(n+1)]^3}{n(n+1)+\frac{[n(n+1)]^3}{n(n+1)+\frac{[n(n+1)]^3}{n(n+1)}+\cdots}}- \frac{n^5(n-1)}{n^2+\frac{n^5(n-1)}{n^2+\frac{n^5(n-1)}{n^2+\cdots}}}$$

$$n^3+n=\frac{[n(n+1)]^3}{n(n+1)+\frac{[n(n+1)]^3}{n(n+1)+\frac{[n(n+1)]^3}{n(n+1)}+\cdots}}- \frac{n^3(n-1)}{n^2+\frac{n^3(n-1)}{n^2+\frac{n^3(n-1)}{n^2+\cdots}}}$$

$$n(n-1)^2=\frac{n^5(n-1)}{n^2+\frac{n^5(n-1)}{n^2+\frac{n^5(n-1)}{n^2+\cdots}}}- \frac{n^3(n-1)}{n^2+\frac{n^3(n-1)}{n^2+\frac{n^3(n-1)}{n^2+\cdots}}}$$

$$n(n+1)^2=\sqrt{[n(n+1)]^3+n(n+1)\sqrt{[n(n+1)]^3+n(n+1)\sqrt{[n(n+1)]^3}+...}}

=n(n+1)+\frac{[n(n+1)]^3}{n(n+1)+\frac{[n(n+1)]^3}{n(n+1)+\cdots}} $$

$$2\phi^2=\frac{\phi^9}{\phi^3+\frac{\phi^9}{\phi^3+\frac{\phi^9}{\phi^3+\cdots}}}-

\frac{\phi^4}{\phi^2+\frac{\phi^4}{\phi^2+\frac{\phi^4}{\phi^2+\cdots}}}$$

$$\left(\frac{\pi}{2\pi+\frac{\pi^2}{6\pi+\frac{\pi^2}{10\pi+\cdots}}}\right)^2= \frac{\frac{e^5(e-1)}{e^2+\frac{e^5(e-1)}{e^2+\frac{e^5(e-1)}{e^2+\cdots}}}- \frac{e^3(e-1)}{e^2+\frac{e^3(e-1)}{e^2+\frac{e^3(e-1)}{e^2+\cdots}}}} {e(e+1)+\frac{[e(e+1)]^3}{e(e+1)+\frac{[e(e+1)]^3}{e(e+1)+\frac{[e(e+1)]^3}{e(e+1)}+\cdots}}}

$$

nth root
$$n=\sqrt[k]{n^{k}-m+\frac{mn}{\sqrt[k]{n^{k}-m+ \frac{mn}{\sqrt[k]{n^{k}-m+\frac{mn}{n^{k}-m+\cdots}}}}}}$$

$$n=\sqrt[k]{n^{k}+m+\frac{mn}{\sqrt[k]{n^{k}+m+ \frac{mn}{\sqrt[k]{n^{k}+m+\frac{mn}{n^{k}+m+\cdots}}}}}}$$

Odd and even
$$x^2-2nx+1=0$$

The solution to this quadratic equation is

$$x=n+\sqrt{n^2-1}=\sqrt{4n^2-1-\frac{2n}{\sqrt{4n^2-1- \frac{2n}{\sqrt{4n^2-1-\frac{2n}{4n^2-1-\cdots}}}}}}$$

$$x^2-(2n+1)x+1=0$$

The solution to this quadratic equation is

$$x=\frac{2n+1+\sqrt{(2n+1)^2-4}}{2}=\sqrt{(2n+1)^2-1-\frac{2n+1}{\sqrt{(2n+1)^2-1- \frac{2n+1}{\sqrt{(2n+1)^2-1-\frac{2n+1}{(2n+1)^2-1-\cdots}}}}}}$$

$$\phi+\sqrt{\phi}=\sqrt{4\phi+3-\frac{2\phi}{\sqrt{4\phi+3- \frac{2\phi}{\sqrt{4\phi+3-\frac{2\phi}{4\phi+3-\cdots}}}}}}$$

$$1=\sqrt{3-\frac{2}{\sqrt{3- \frac{2}{\sqrt{3-\frac{2}{3-\cdots}}}}}}$$

$$\phi^2=\sqrt{8-\frac{3}{\sqrt{8- \frac{3}{\sqrt{8-\frac{3}{8-\cdots}}}}}}$$

$$\phi^4=\sqrt{48-\frac{7}{\sqrt{48- \frac{7}{\sqrt{48-\frac{7}{48-\cdots}}}}}}$$

$$2=\sqrt{3+\frac{2}{\sqrt{3+ \frac{2}{\sqrt{3+\frac{2}{3+\cdots}}}}}}$$

$$\phi=\sqrt{2+\frac{1}{\sqrt{2+ \frac{1}{\sqrt{2+\frac{1}{2+\cdots}}}}}}$$

$$1=\sqrt{2-\frac{1}{\sqrt{2- \frac{1}{\sqrt{2-\frac{1}{2-\cdots}}}}}}$$

$$2\phi=\sqrt{8+\frac{8}{\sqrt{8+ \frac{8}{\sqrt{8+\frac{8}{8+\cdots}}}}}}$$

$$2=\sqrt{8-\frac{8}{\sqrt{8- \frac{8}{\sqrt{8-\frac{8}{8-\cdots}}}}}}$$

$$\sqrt{2+\sqrt{3}}=\sqrt{3+\frac{\sqrt{2}}{\sqrt{3+ \frac{\sqrt{2}}{\sqrt{3+\frac{\sqrt{2}}{3+\cdots}}}}}}$$

$$\sqrt{2}=\sqrt{3-\frac{\sqrt{2}}{\sqrt{3- \frac{\sqrt{2}}{\sqrt{3-\frac{\sqrt{2}}{3-\cdots}}}}}}$$

$$\sqrt{3}=\sqrt{2+\frac{\sqrt{3}}{\sqrt{2+ \frac{\sqrt{3}}{\sqrt{2+\frac{\sqrt{3}}{2+\cdots}}}}}}$$

$$\sqrt{k+1}=\sqrt{k+\frac{\sqrt{k+1}}{\sqrt{k+ \frac{\sqrt{k+1}}{\sqrt{k+\frac{\sqrt{k+1}}{k+\cdots}}}}}}$$

$$\sqrt{k-1}=\sqrt{k-\frac{\sqrt{k-1}}{\sqrt{k- \frac{\sqrt{k-1}}{\sqrt{k-\frac{\sqrt{k-1}}{k-\cdots}}}}}}$$

Relation of pi and e
$$\frac{\pi}{2\pi+\frac{\pi^2}{6\pi+\frac{\pi^2}{10\pi+\cdots}}}=\frac{e-\frac{\sqrt{e-1}}{\sqrt{e- \frac{\sqrt{e-1}}{\sqrt{e-\frac{\sqrt{e-1}}{e-\cdots}}}}}}{e+\frac{\sqrt{e+1}}{\sqrt{e+ \frac{\sqrt{e+1}}{\sqrt{e+\frac{\sqrt{e+1}}{e+\cdots}}}}}}$$

$$1=\sqrt{\phi^2-\frac{\phi}{\sqrt{\phi^2- \frac{\phi}{\sqrt{\phi^2-\frac{\phi}{\phi^2-\cdots}}}}}}$$

$$\phi=\sqrt{\phi^3-\frac{\phi^2}{\sqrt{\phi^3- \frac{\phi^2}{\sqrt{\phi^3-\frac{\phi^2}{\phi^3-\cdots}}}}}}$$

$$1+\sqrt{k+1}=\sqrt{2^2+k+\frac{2k}{\sqrt{2^2+k+ \frac{2k}{\sqrt{2^2+k+\frac{2k}{2^2+k+\cdots}}}}}}$$

$$\frac{e^{\frac{2y}{x}}-1} {e^{\frac{2y}{x}}+1}=\frac{y}{1x+\frac{y^2}{3x+\frac{y^2}{5x+\cdots}}}=\frac{e^{\frac{2y}{x}}- \frac{\sqrt{e^{\frac{2y}{x}}-1}}{\sqrt{e^{\frac{2y}{x}}- \frac{\sqrt{e^{\frac{2y}{x}}-1}}{\sqrt{e^{\frac{2y}{x}}- \frac{\sqrt{e^{\frac{2y}{x}}-1}}{e^{\frac{2y}{x}}- \cdots}}}}}}{e^{\frac{2y}{x}}+\frac{\sqrt{e^{\frac{2y}{x}}+1}} {\sqrt{e^{\frac{2y}{x}}+ \frac{\sqrt{e^{\frac{2y}{x}}+1}}{\sqrt{e^{\frac{2y}{x}}+ \frac{\sqrt{e^{\frac{2y}{x}}+1}}{e^{\frac{2y}{x}}+\cdots}}}}}}$$

Power of 2
$$2^n=2^{n-1}+\frac{2^{2n-1}}{2^{n-1}+\frac{2^{2n-1}}{2^{n-1}+\frac{2^{2n-1}}{2^{n-1}+\cdots}}}$$

Special conditions

n = 2

$$2^2=2+\frac{2^3}{2+\frac{2^3}{2+\frac{2^3}{2+\cdots}}}$$

n = 1

$$2=1+\frac{2}{1+\frac{2}{1+\frac{2}{1+\cdots}}}$$

n = 0

$$1=\frac{1}{2}+\frac{\frac{1}{2}}{\frac{1}{2}+\frac{\frac{1}{2}}{\frac{1}{2}+ \frac{\frac{1}{2}}{\frac{1}{2}+\cdots}}}$$

n = 1/2

$$\sqrt{2}=\frac{1}{\sqrt{2}}+\frac{1}{\frac{1}{\sqrt{2}}+ \frac{1}{\frac{1}{\sqrt{2}}+\frac{1}{\frac{1}{\sqrt{2}}+\cdots}}}$$

Others

$$\frac{\sqrt{4^4+4^2}+4^1}{2}=4+\frac{4^3}{4+\frac{4^3}{4+\frac{4^3}{4+\cdots}}}$$

Phi
$$\frac{2\phi+1}{2}=2+\frac{1}{8+\frac{1}{2+\frac{1}{8+\cdots}}}$$

$$\sqrt{\phi}=z+\frac{1}{z+\frac{1}{z+\frac{1}{z+\cdots}}}$$

Where

$$z=\frac{1}{\phi\sqrt{\phi}}$$

$$k\phi=k+\frac{1}{k^{-1}+\frac{1}{k+\frac{1}{k^{-1}+\frac{1}{k+\cdots}}}}$$

k = 1

$$\phi=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}}$$

k = 2

$$2\phi=2+\frac{1}{0.5+\frac{1}{2+\frac{1}{0.5+\frac{1}{2+\cdots}}}}$$

$$k=\phi$$

$$1=\frac{1}{\phi}+\frac{1}{\phi+\frac{1}{\frac{1}{\phi}+\frac{1}{\phi+\frac{1}{\frac{1}{\phi}+\cdots}}}}$$

Others

$$4=2+\frac{1}{0.25+\frac{1}{2+\frac{1}{0.25+\frac{1}{2+\cdots}}}}$$

$$4=2+\frac{2}{0.5+\frac{2}{2+\frac{2}{0.5+\frac{2}{2+\cdots}}}}$$

Phi and ...
$$\sqrt{1+\frac{1}{n}}=1+\frac{1}{\sqrt{n(n+1)}+\frac {\sqrt{n(n+1)}}{1+\frac{1}{\sqrt{n(n+1)}+\cdots}}}$$

$$n=\phi$$

$$\sqrt{\phi}=1+\frac{1}{\sqrt{\phi^3}+\frac{\sqrt{\phi^3}}{1+\frac{1}{\sqrt{\phi^3}+\cdots}}}$$

$$\phi=\frac{\sqrt{\phi^3}}{1+\frac{1}{\sqrt{\phi^3}+\frac{\sqrt{\phi^3}}{1+\cdots}}}$$

$$\sqrt{\phi}=1+\frac{1}{\sqrt{\phi^3}+\phi}$$

$$n=\phi^3$$

$$\sqrt{\frac{2}{\phi}}=1+\frac{1}{\sqrt{2\phi^5}+\frac{\sqrt{2\phi^5}}{1+\frac{1}{\sqrt{2\phi^5}+\cdots}}}$$

$$\phi^3=\frac{\sqrt{2\phi^5}}{1+\frac{1}{\sqrt{2\phi^5}+\frac{\sqrt{2\phi^5}}{1+\cdots}}}$$

$$\sqrt{\frac{2}{\phi}}=1+\frac{1}{\sqrt{2\phi^5}+\phi^3}$$

$$n=\phi^4$$

$$\sqrt{\frac{3}{\phi^2}}=1+\frac{1}{\sqrt{3\phi^6}+\frac{\sqrt{3\phi^6}}{1+\frac{1} {\sqrt{3\phi^6}+\cdots}}}$$

$$\phi^4=\frac{\sqrt{3\phi^6}}{1+\frac{1}{\sqrt{3\phi^6}+\frac{\sqrt{3\phi^6}}{1+\cdots}}}$$

$$\sqrt{\frac{3}{\phi^2}}=1+\frac{1}{\sqrt{3\phi^6}+\phi^4}$$

Unity and phi
$$\sqrt{X}=1+\frac{1}{\sqrt{Y}+Z}$$

$$Z=\frac{\sqrt{Y}}{1+\frac{1}{\sqrt{Y}+\frac{\sqrt{Y}}{1+\cdots}}}$$

$$\sqrt{\frac{\phi{F_{2n-1}}}{F_{2n}}}= 1+\frac{1}{\sqrt{\phi^{4n-1}F_{2n-1}F_{2n}}+\phi^{2n-1}F_{2n}}$$

$$\sqrt{\frac{\phi}{1}}=1+\frac{1}{\sqrt{\phi^3}+\phi}$$

$$\sqrt{\frac{2\phi}{3}}=1+\frac{1}{\sqrt{6\phi^7}+3\phi^3}$$

$$\sqrt{\frac{5\phi}{8}}=1+\frac{1}{\sqrt{40\phi^{11}}+8\phi^5}$$

$$\sqrt{\frac{F_{2n+1}}{\phi{F_{2n}}}}=1+\frac{1}{\sqrt{\phi^{4n+1}F_{2n+1}F_{2n}}+ \phi^{2n+1}F_{2n}}$$

$$\sqrt{\frac{2}{\phi}}=1+\frac{1}{\sqrt{2\phi^5}+\phi^3}$$

$$\sqrt{\frac{5}{3\phi}}=1+\frac{1}{\sqrt{15\phi^9}+3\phi^5}$$

$$\sqrt{\frac{13}{8\phi}}=1+\frac{1}{\sqrt{104\phi^{13}}+8\phi^7}$$

Others

$$\sqrt{\frac{F_{2n+2}}{\phi^2{F_{2n}}}}=1+\frac{1}{\sqrt{\phi^{4n+2}F_{2n+2}F_{2n}}+ \phi^{2n+2}F_{2n}}$$

$$\sqrt{\frac{3}{\phi^2}}=1+\frac{1}{\sqrt{3\phi^6}+\phi^4}$$

$$\sqrt{\frac{8}{3\phi^2}}=1+\frac{1}{\sqrt{24\phi^{10}}+3\phi^6}$$

$$\sqrt{\frac{21}{8\phi^2}}=1+\frac{1}{\sqrt{168\phi^{14}}+8\phi^8}$$

$$\sqrt{\frac{\phi^2F_{2n-1}}}=1+\frac{1}{\sqrt{\phi^{4n}F_{2n+1}F_{2n-1}}+ \phi^{2n-1}F_{2n+1}}$$

$$\sqrt{\frac{\phi^2}{2}}=1+\frac{1}{\sqrt{2\phi^4}+2\phi^1}$$

$$\sqrt{\frac{2\phi^2}{5}}=1+\frac{1}{\sqrt{10\phi^8}+5\phi^3}$$

$$\sqrt{\frac{5\phi^2}{13}}=1+\frac{1}{\sqrt{65\phi^{12}}+13\phi^5}$$

Others

$$\sqrt{2}=\left(1+\frac{1}{\sqrt{\phi^3}+\phi}\right) \left(1+\frac{1}{\sqrt{2\phi^5}+\phi^3}\right)$$

Lucas number
$$\frac{\phi^{2n-1}}{L_{2n}}=1+\frac{1}{\phi^{2(2n-1)}-1}$$

$$\frac{\phi}{1}=1+\frac{1}{\phi^2-1}$$

$$\frac{\phi^3}{4}=1+\frac{1}{\phi^6-1}$$

others

$$\frac{\phi^2}{2}=1+\frac{1}{\phi^3-1}$$

$$\frac{\phi^2}{\sqrt{5}}=1+\frac{1}{\phi^4-1}$$

$$\frac{\phi^3}{\phi^4-3}=1+\frac{1}{\phi^5-1}$$

$$\frac{\phi^4}{\phi+5}=\frac{\phi^3}{\phi^5-7}=1+\frac{1}{\phi^7-1}$$

Ramanujan
Here I plays around with Ramanujan's equation and I got this:

$$\frac{1}{\sqrt{2\phi^4-4\phi^3+3\phi^2-2\phi^1}-\phi^0}= \left(1+\frac{e^{-2\pi}}{1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}{1+\cdots}}}\right) \left({e^{\frac{2\pi}{5}}+\frac{1}{e^{-\frac{2\pi}{5}}+\frac{1} {e^{\frac{2\pi}{5}}+\frac{1}{e^{-\frac{2\pi}{5}}+\cdots}}}}\right)$$

$$ \frac{1}{\sqrt{13\phi^8-29\phi^7+18\phi^6-11\phi^5+7\phi^4-4\phi^3+3\phi^2-5\phi}+\phi^3-3\phi^2+2\phi} = \left(1-\frac{e^{-\pi}}{1+\frac{e^{-2\pi}} {1-\frac{e^{-3\pi}}{1+\cdots}}}\right) \left({e^{\frac{\pi}{5}}+\frac{1}{e^{-\frac{\pi}{5}}+\frac{1} {e^{\frac{\pi}{5}}+\frac{1}{e^{-\frac{\pi}{5}}+\cdots}}}}\right)$$

$$y=3-\phi$$

$$x=2+\phi$$

$$\frac{e^{-\frac{2\pi}{5}}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}{1+\cdots}}}} =\sqrt{y}+\frac{y}{\sqrt{y}+\frac{y}{\sqrt{y}+\frac{y}{\sqrt{y}+\cdots}}} -\left(1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}\right)$$

$$\frac{e^{-\frac{\pi}{5}}}{1-\frac{e^{-\pi}}{1+\frac{e^{-2\pi}} {1-\frac{e^{-3\pi}}{1+\cdots}}}} =\frac{x}{\sqrt{x}+\frac{x}{\sqrt{x}+\frac{x}{\sqrt{x}+\cdots}}} -\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}$$

Lucas numbers

$$\frac{\sqrt{5}}{1+\sqrt[5]{5^{\frac{3}{4}}(\phi-1)-1}}-\phi=\frac{e^{-\frac{2\pi}{\sqrt{5}}}} {1+\frac{e^{2\phi^0{\pi}(1-2\phi)}}{1+\frac{e^{4\phi^2{\pi}(4-3\phi)}} {1+\frac{e^{6\phi^4{\pi}(11-7\phi)}}{1+\frac{e^{8\phi^6{\pi}(29-18\phi)}}{1+\cdots}}}}}$$

Power of phi
$$2\phi^n=L_{n+1}+\sqrt{5}F_n$$

Others

$$\sum_{n=1}^{M+1}F^2_n=\phi^M{F_{M+3}}-\phi^{2M}-\frac{F^2_{M}}{\phi^3}$$

$$n=\sqrt{n(n+1)}+\frac{1}{1+\frac{\sqrt{n(n+1)}}{\sqrt{n(n+1)}+\frac{1}{1+\cdots}}}$$

$$\phi=\sqrt{\phi^3}-\frac{1}{1+\sqrt{\phi}}$$

$$2\phi=\phi^2\sqrt{2}-\frac{1}{1+\frac{\phi}{\sqrt{2}}}$$

$$\phi^3=\sqrt{2\phi^5}-\frac{1}{1+\sqrt{\frac{2}{\phi}}}$$

$$\phi^{n+2}F_n-\phi^nF_{n+2}=(-1)^{n+1}$$

$$\sqrt{5}+\frac{1}{1+\sqrt{\frac{1}{\sqrt{5}}+1}}=\sqrt{2\phi+4}$$

$$F_m{\phi^n}=\phi^m{F_n}+(-1)^{m+1}F_{n-m}$$

Special case

m = 1

$$\phi^n=\phi{F_n}+F_{n-1}$$

$$1+\frac{k}{1+\frac{k^{-1}}{1+\frac{k}{1+\frac{k^{-1}}{1+\cdots}}}}= k\times{\left(k^{-1}+\frac{k}{k+\frac{k^{-1}}{k^{-1}+\frac{k}{k+ \frac{k^{-1}}{k^{-1}+\cdots}}}}\right)}$$

$$k+\frac{k^{-1}}{k+\frac{k}{k+\frac{k^{-1}}{k+\frac{k}{k+\cdots}}}}= k\times{\left(1+\frac{k^{-1}}{k^2+\frac{k}{1+\frac{k^{-1}}{k^2+ \frac{k}{1+\cdots}}}}\right)}$$

$$k+\frac{k}{k^{-1}+\frac{k^{-1}}{k+\frac{k}{k^{-1}+\frac{k^{-1}}{k+\cdots}}}}= k\times{\left(1+\frac{k}{1+\frac{k^{-1}}{1+\frac{k}{1+ \frac{k^{-1}}{1+\cdots}}}}\right)}$$

$$k^n+\frac{k}{k^{-n}+\frac{k^{-1}}{k^n+\frac{k}{k^{-n}+\frac{k^{-1}}{k^n+\cdots}}}}= k^{n+1}\times{\left(k^{-1}+\frac{k}{k+\frac{k^{-1}}{k^{-1}+\frac{k}{k+ \frac{k^{-1}}{k^{-1}+\cdots}}}}\right)}$$

$$\sqrt{2}+1=1+\frac{1}{2^{-1}+\frac{2^{-1}}{1+\frac{1}{2^{-1}+\frac{2^{-1}}{1+\cdots}}}}$$

$$\phi^3=1+\frac{1}{4^{-1}+\frac{4^{-1}}{1+\frac{1}{4^{-1}+\frac{4^{-1}}{1+\cdots}}}}$$

$$\frac{1}{5\phi}+8=1+\frac{1}{8^{-1}+\frac{8^{-1}}{1+\frac{1}{8^{-1}+\frac{8^{-1}}{1+\cdots}}}}$$