User:Proof2013

user:proof2012

Phi
$$arctan x$$

$$F_1=1$$, $$F_2=1$$ then $$F_{n+2}=F_{n}+F_{n+1}$$

$$L_1=2$$, $$L_2=1$$ then $$L_{n+2}=L_{n}+L_{n+1}$$

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

$$\left(-\phi^{-1}\right)^n=\frac{L_{n+1}-\sqrt{5}F_n}{2}$$

$$(-1)^n4=L_{n+1}^2-5F_{n}^2$$

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

$$\left(-\phi^2\right)^n=\frac{L_{n+1}+\sqrt{5}F_n}{L_{n+1}-\sqrt{5}F_n}$$

$$2\phi^nF_k+F_n\left[\left(-\phi^{-1}\right)^k-\phi^k\right]=F_{n+k}+F_{n-k}$$

$$2\phi^nF_k+(-1)^k\sqrt{5}F_kF_n=F_{n+k}+F_{n-k}$$

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

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

$$\phi^nF_k+\left(-\phi^{-1}\right)^{k}F_{n}=F_{n+k}$$

$$L_1=2$$, $$L_2=1$$

$$L_{n+2}=L_n+F_{n+1}$$

$$\phi^kF_{2n}+\phi^{2n}F_{k}+\frac{2F_k}{\phi^{2n}}=L_kF_{2n-1}+F_{k+2n+1}$$

$$\phi^kF_{2n-1}+\phi^{2n-1}F_{k}-\frac{2F_k}{\phi^{2n-1}}=L_kF_{2n-2}+F_{k+2n}$$

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Let

$$L_1=1$$, $$L_2=3$$,

$$L_{n+2}=L_n+L_{n+1}$$

$$N=1,2,3,\cdots$$

$$N_o=1,3,5,\cdots$$

$$N_e=2,4,6,\cdots$$

$$\phi^{N_oA}+(-1)^A\phi^{-N_oA}=\frac{(-1)^{A+1}F_{(N_o-1)A}+F_{(N_o+1)A}}{F_{A}}$$

$$\phi^{N_eA}+\phi^{-N_eA}=\frac{(-1)^{A+1}F_{(N_e-1)A}+F_{(N_e+1)A}}{F_{A}}$$

$$L_{nA}=\frac{(-1)^{A+1}F_{(N-1)A}+F_{(N+1)A}}{F_{A}}$$