User:Limitphi

user:continuedfraction

Others
 _n  : 1   2   3   4   5   6   7  ...

Fn : 1   1   2   3   5   8   13 ...

Ln : 1 3 4 7 11...

$$\sum_{k=1}^{2n}L_{k}L_{x+k-1}=5F_{2n}F_{2n+x}$$

$$\sum_{k=1}^{2n}F_{k}F_{x+k-1}=F_{2n}F_{2n+x}$$

$$F_{n+1}^3+F_{n+2}^3-F_{n}^3=F_{3n+3}$$

$$\frac{L_n^2-L_{n+1}^2+L_{n+2}^2}{F_n^2-F_{n+1}^2+F_{n+2}^2}= \frac{L_nL_{n+2}}{F_nF_{n+2}}$$

5
$$\frac{\sum_{k=1}^{2n}L_{k}L_{x+k-1}}{\sum_{k=1}^{2n}F_{k}F_{x+k-1}}=5$$

m = {1,2,3,...}

a = {1,3,5,...}

$$\frac{\sum_{k=1}^{2m}L_{n+a(k-1)}^2}{\sum_{k=1}^{2m}F_{n+a(k-1)}^2}=5$$

let m = 2 we have,

$$\frac{L_n^2+L_{n+a}^2+L_{n+2a}^2+L_{n+3a}^2}{F_n^2+F_{n+a}^2+F_{n+2a}^2+F_{n+3a}^2}=5$$

Odd
$$\frac{ -4(-1)^n+\sum_{k=1}^{2m-1}L_{n+a(k-1)}^2 }{ \sum_{k=1}^{2m-1}F_{n+a(k-1)}^2 }=5$$

Let m = 1 we have,

$$\frac{L_n^2-4(-1)^n}{F_n^2}=5$$

Let m = 2 we have,

$$\frac{L_n^2+L_{n+a}^2+L_{n+2a}^2-4(-1)^n}{F_n^2+F_{n+a}^2+F_{n+2a}^2}=5$$

and so on ...

Even
b = {0,2,4,6,...}

$$\frac{-8m(-1)^n+\sum_{k=1}^{2m}L_{n+b(k-1)}^2}{\sum_{k=1}^{2m}F_{n+b(k-1)}^2}=5$$

Let m = 1 we have,

$$\frac{L_n^2+L_{n+b}^2-8(-1)^n}{F_n^2+F_{n+b}^2}=5$$

Let m = 2 we have,

$$\frac{L_n^2+L_{n+b}^2+L_{n+2b}^2+L_{n+3b}^2-16(-1)^n}{F_n^2+F_{n+b}^2+F_{n+2b}^2+F_{n+3b}^2}=5$$

Definition
 m/n  : 1   2   3   4   5   6   7  ...

Fn : 1   1   2   3   5   8   13 ...

Ln : 2 1 3 4 7 11...

Main
$$Z_k=\frac{\left(\frac{L_n+\sqrt{L_n^2+4L_n}}{2}\right)^k-\left(\frac{L_n-\sqrt{L_n^2+4L_n}}{2}\right)^k}{\sqrt{L_n^2+4L_n}}$$

$$Z_1^2+Z_2^2+Z_3^2+Z_4^2+\cdots+Z_k^2=\frac{Z_kZ_{k+1}}{L_n}$$

$$\sum_{k=1}^{m}\left(\frac{L_{k}}{L_{k+1}L_{k+2}}-\frac{F_{k}}{F_{k+1}F_{k+2}}\right)=\frac{F_{m}}{F_{2m+3}-(-1)^m}= \frac{1}{F_{m+1}}-\frac{1}{L_{m+2}}$$

$$5F_nL_{n+1}+10F_{2n+1}=L_{n+2}^2+L_{n+3}^3$$

$$F_nF_{n+1}+F_{n+2}F_{n+3}+L_{2n+5}=F_{n+4}^2$$

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

Analogue

$$\phi^{n+1}-\phi^{n-3}=\phi{L_{n}}+L_{n-1}$$

$$\phi^n+2\phi^{n-2}-\phi(F_{n+1}+F_{n-4})=F_{n}+F_{n-5}$$

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

n > 3

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

$$L_{2n}L_{2n+2}L_{2n+4}L_{2n+6}+5^3F_{2n+2}^2=L_{2n+3}^4$$

$$F_{2n}F_{2n+2}F_{2n+4}F_{2n+6}+L_{2n+4}^2=F_{2n+3}^4$$

$$L_{2n}L_{2n+2m}+5F_{2m}^2=L_{2n+m}^2$$

$$F_{2n}F_{2n+2m}+F_{2m}^2=F_{2n+m}^2$$

$$L_{4n+2}L_{4n+6}+5=L_{4n+4}^2$$

$$F_{4n+2}F_{4n+6}+1=F_{4n+4}^2$$

$$\frac{\phi^6+14}{\phi^6-4\phi}=\frac{3^3\phi+2^3}{3^2\phi+2^2}$$

$$2F_{n-3}=\phi^3{F_{n-4}}+\phi^2{L_{n-3}}-\phi{F_{n-1}}$$

Phi
$$\phi^n=\phi^3{F_{n-4}}+\phi^2{L_{n-3}}+\phi{F_{n-2}}+F_{n-4}$$

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

$$F_{2n-1}=3F_{n}^2+2F_{n+1}^2-F_{n+2}^2$$

$$L_{n+1}^2=3F_{n}^2+6F_{n+1}^2-2F_{n+2}^2$$

$$L_{2n+1}=4F_{n}^2+3F_{n+1}^2-F_{n+2}^2$$

Sum
$$\sum_{i=1}^{2nk}F_{i}^2=F_{2n}\sum_{i=1}^{k}F_{4ni-2n+1}$$

$$\sum_{k=0}^{2m-1}F_{n+k}^2=F_{2m}F_{2n+2m-1}$$

$$\sum_{i=1}^{4n}F_i^2=3\sum_{i=1}^{n}F_{8i-3}$$

$$5\sum_{i=1}^{n}F_{i}^2=F_{n+3}^2-F_{n+2}^2-F_{n+1}^2-F_{n+1}$$

$$3\sum_{i=1}^{n}F_{i}^2=-F_{n+1}^2+F_{n+2}^2+F_{n-1}F_n$$

$$2\sum_{i=1}^{n}F_{i}^2=-F_{n-1}^2+F_{n}^2+F_{n+1}^2$$

$$F_{n}^2-3F_{n+1}^2+F_{n+2}^2=2(-1)^{n+1}$$

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

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

$$F_n{\times}\frac{F_{n+2}^3+F_{n+3}^3}{F_{n+4}}+F_{n+1}^3+F_{n+2}^3= \left(F_{n}+F_{n+2}\right){\times}\frac{F_{n+1}^3+F_{n+2}^3}{F_{n+3}}$$

$$\frac{5+3\sqrt{5}}{5}=\lim_{n\to \infty}F_n^2\phi^3-F_{n+1}^2\phi$$

$$\phi^k=\lim_{n\to\infty}\frac{F_{n+1}^k+F_{n+2}^k}{F_n^k+F_{n+1}^k}$$