Fibbonacci
fibbonacci¶
\[ F_n = F_{n - 1} + F_{n - 2} \]
\[ F_1 = 1 \]
\[ F_2 = 1 \]
\[ x^2 = x + 1 \]
\[ x = \begin{Bmatrix} \frac{1}{2} + \frac{\sqrt{5}}{2} = \phi \\
\frac{1}{2} - \frac{\sqrt{5}}{2} = \psi \\ \end{Bmatrix} \]
\[ (-\frac{1}{x})^2 = \frac{1}{x^2} \]
\[ 1 = \frac{1}{x} + \frac{1}{x^2} \]
\[ 1 - \frac{1}{x} = \frac{1}{x^2} = (-\frac{1}{x})^2 \]
\[ (-\frac{1}{x})^2 = (-\frac{1}{x}) + 1 \]
\[ x = 1 + \frac{1}{x} \]
\[ -\frac{1}{x} ? = ? x = 1 + \frac{1}{x} \]
\[ -1 - \frac{1}{x} = -x ? = ? \frac{1}{x} \]
\[ -x^2 ? = ? 1 \]
\[ x ? = ? i, -i \]
\[ \phi \ne i \]
\[ \phi \ne -i \]
\[ \psi \ne i \]
\[ \psi \ne -i \]
\[ \text{Thus...} \]
\[ \phi^{-1} = -\psi \]
\[ \psi^{-1} = -\phi \]
\[ \psi^{n - 1} = -\phi \psi^n \]
\[ \phi^{n - 1} = -\psi \phi^n \]
\[ \text{And don't forget that!} \]
\[ \phi^2 = \phi + 1 \]
\[ \phi^3 = \phi^2 + \phi = 2 \phi + 1 \]
\[ \phi^4 = 2 \phi^2 + \phi = 3 \phi + 2 \]
\[ \vdots \]
\[ \phi^n = c_{n, n} \phi + c_{n, n - 1} \]
\[ \phi^{n + 1} = c_{n + 1, n + 1} \phi + c_{n + 1, n} = \phi^n \phi = c_{n, n} \phi^2 + c_{n, n - 1} \phi = c_{n, n} \phi + c_{n, n} + c_{n, n - 1} \phi \]
\[ c_{n + 1, n} = c_{n, n} \]
\[ c_{n + 2, n} = c_{n + 1, n} = c_{n, n} \]
\[ \vdots \]
\[ c_{n + k, n} = c_{n, n} \]
\[ C_n = : c_{n, n} \]
\[ \phi^n = C_n \phi + C_{n - 1} \]
\[ \phi^{n + 1} = C_{n + 1} \phi + C_n = \phi^n \phi = C_n \phi^2 + C_{n - 1} \phi = C_n \phi + C_n + C_{n - 1} \phi \]
\[ C_{n + 1} = C_n + C_{n - 1} \]
\[ C_n = C_{n - 1} + C_{n - 2} \]
\[ \phi^2 = \phi + 1 = \phi C_2 + C_1 \]
\[ C_1 = 1 \]
\[ C_2 = 1 \]
\[ C_n = F_n \]
\[ \phi^n = F_n \phi + F_{n - 1} \]
\[ \text{Yay, now I can solve for $F_n$ if it weren't for the second fibbonacci term, so how can I solve that?} \]
\[ \text{Well, this is only because $\phi^2 = \phi + 1$, but same goes for $\psi$, so...} \]
\[ \psi^n = F_n \psi + F_{n - 1} \]
\[ \text{And subtracting, we get...} \]
\[ \phi^n - \psi^n = F_n \phi + F_{n - 1} - F_n \psi - F_{n - 1} = (\phi - \psi) F_n = (\frac{1}{2} + \frac{\sqrt{5}}{2} - \frac{1}{2} + \frac{\sqrt{5}}{2}) F_n = \sqrt{5} F_n \]
\[ F_n = \frac{\phi^n - \psi^n}{\sqrt{5}} = \frac{\phi^n - \psi^n}{\phi - \psi} \]
\[ F_n \approx \frac{\phi^n}{\sqrt{5}} \]
luca numbers¶
\[ L_n = : F_{n + 1} + F_{n - 1} = L_{n - 1} + L_{n - 2} \]
\[ L_1 = 1 \]
\[ L_2 = 3 \]
\[ L_0 = 2 \text{ And } F_0 = 0 \text{ btw.} \]
\[ L_n = \frac{\phi^{n + 1} - \psi^{n + 1}}{\phi - \psi} + \frac{\phi^{n - 1} - \psi^{n - 1}}{\phi - \psi} = \frac{\phi^{n + 1} - \psi^{n + 1} + \phi^{n - 1} - \psi^{n - 1}}{\phi - \psi} \]
\[ \text{I hope that you remembered!} \]
\[ L_n = \frac{\phi^n \phi - \psi^n \psi - \phi^n \psi + \psi^n \phi}{\phi - \psi} = \frac{\phi^n (\phi - \psi) + \psi^n (\phi - \psi)}{\phi - \psi} \]
\[ L_n = \phi^n + \psi^n \]
\[ L_n \approx \phi^n \]
fibbonacci numbers?¶
\[ L_{n + 1} + L_{n - 1} = \phi^{n + 1} + \psi^{n + 1} + \phi^{n - 1} + \psi^{n - 1} = \phi^n \phi + \psi^n \psi - \phi^n \psi - \psi^n \phi = \phi^n (\phi - \psi) - \psi^n (\phi - \psi) = \phi^n \sqrt{5} - \psi^n \sqrt{5} = \frac{\phi^n - \psi^n}{\sqrt{5}} 5 \]
\[ L_{n + 1} + L_{n - 1} = 5 F_n \]
\[ F_n = \frac{L_{n + 1} + L_{n - 1}}{5} \]
\[ L_n \phi + L_{n - 1} = (\phi^n + \psi^n) \phi + \phi^{n - 1} + \psi^{n - 1} = \phi^{n + 1} + \psi^n \phi + \phi^{n - 1} - \psi^n \phi = \phi^{n + 1} + \phi^{n - 1} = \phi^n (\phi + \phi^{-1}) \]
\[ \phi^n \sqrt{5} = L_n \phi + L_{n - 1} = \phi^{n + 1} + \phi^{n - 1} \]
\[ \phi^n = \frac{L_n \phi + L_{n - 1}}{\sqrt{5}} = \frac{\phi^{n + 1} + \phi^{n - 1}}{\sqrt{5}} \]
\[ \phi^n = \phi^{n - 1} + \phi^{n - 2} \text{ btw} \]
everything up 'til now¶
\[ F_1 = 1 \]
\[ F_2 = 1 \]
\[ F_0 = 0 \]
\[ \phi^1 = \phi \]
\[ \phi^2 = \phi + 1 \]
\[ \phi^0 = 1 \]
\[ L_1 = 1 \]
\[ L_2 = 3 \]
\[ L_0 = 2 \]
.
\[ F_n = F_{n - 1} + F_{n - 2} \]
\[ \phi^n = \phi^{n - 1} + \phi^{n - 2} \]
\[ L_n = L_{n - 1} + L_{n - 2} \]
.
\[ F_n = \frac{\phi^n - \psi^n}{\sqrt{5}} \]
\[ \phi^n = F_n \phi + F_{n - 1} = \frac{L_n \phi + L_{n - 1}}{\sqrt{5}} \]
\[ L_n = \phi^n + \psi^n \]
.
\[ F_n = \frac{L_{n + 1} + L_{n - 1}}{5} \]
\[ \phi^n = \frac{\phi^{n + 1} + \phi^{n - 1}}{\sqrt{5}} \]
\[ L_n = F_{n + 1} + F_{n - 1} \]
.
\[ F_n \approx \frac{\phi^n}{\sqrt{5}} \]
\[ \phi^n \approx \frac{F_{n + 1}}{F_n} \]
\[ L_n \approx \phi^n \]
one more thing¶
\[ \text{how would one combine } \phi^n = F_n \phi + F_{n - 1} \text{ and } \phi^n = \frac{L_n \phi + L_{n - 1}}{\sqrt{5}}? \]
\[ \text{for one, rearrange the second one} \]
\[ \phi^n = F_n \phi + F_{n - 1} \]
\[ \phi^n \sqrt{5} = L_n \phi + L_{n - 1} \]
\[ F_{n - 1} = F_n - F_{n - 2} \]
\[ L_{n - 1} = F_n + F_{n - 2} \]
\[ \phi^n = F_n \phi + F_n - F_{n - 2} \]
\[ \phi^n \sqrt{5} = L_n \phi + F_n + F_{n - 2} \]
\[ (1 + \sqrt{5}) \phi^n = 2 \phi \phi^n = (F_n + L_n) \phi + F_n - F_{n - 2} + F_n + F_{n - 2} = (F_n + L_n) \phi + 2F_n \]
\[ \phi \phi^n = \frac{1}{2} (F_n + L_n) \phi + F_n \]
\[ \phi^n = \frac{1}{2} (F_n + L_n) + F_n \phi^{-1} = \frac{1}{2} F_n + \frac{1}{2} L_n - F_n \psi = \frac{1}{2} F_n + \frac{1}{2} L_n - \frac{1}{2} F_n + \frac{\sqrt{5}}{2} F_n \]
\[ \phi^n = \frac{1}{2} L_n + \frac{\sqrt{5}}{2} F_n \approx \frac{1}{2} \phi^n + \frac{\sqrt{5}}{2} \frac{\phi^n}{\sqrt{5}} = \phi^n \]
fibbonacci 2¶
\[ S_n = ? \]
\[ S_0 = a \]
\[ S_1 = b \]
\[ S_n = S_{n - 1} + S_{n - 2} \]
\[ F_{-1} = 1 \text{ Btw.} \]
\[ S_2 = a + b \]
\[ S_3 = a + b + b = a + 2b \]
\[ S_4 = a + 2b + a + b = 2a + 3b \]
\[ S_5 = 2a + 3b + a + 2b = 3a + 5b \]
Yes, I know that this text is not centered and is in a different font. I'm tired of proving things that the corrasponding youtube narrator just says "I'll leave this as an exersize for the veiwer", for example I used lines \(58\) thru \(88\) just to prove that the numbers that you were seeing were the fibbonacci numbers, I got too lazy to prove this next one, so I hate to say it, but I will leave proving the following statement as an exersize for the viewer:
\[ S_n = F_n b + F_{n - 1} a = \frac{(\phi^n - \psi^n) b + (\phi^{n - 1} - \psi^{n - 1}) a}{\phi - \psi} = \frac{\phi^n b - \psi^n b - \psi \phi^n a + \phi \psi^n a}{\phi - \psi} \]
\[ S_n = \frac{\phi^n (b - \psi a) - \psi^n (b - \phi a)}{\phi - \psi} \]