(41) Fibonacci & Lucas numbers

[Albert Chan]

Also from the book, Number Theory in Science and Communication, allowed complex numbers, we have:

Let \(\large z_1 = \cos^{-1}({-i \over 2})\quad\;
→ F_n = i^{n-1} {\sin (n z_1) \over \sin (z_1)} \quad\;
→ L_n = i^{n-1} {\cos (n z_1) \over \cos(z_1)} = 2\; i^n \cos (n z_1) \)

Or, with hyperbolics:

Let \(\large z_2 = \cosh^{-1}({-i \over 2})\quad
→ F_n = i^{n-1} {\sinh (n z_2) \over \sinh (z_2)} \quad
→ L_n = i^{n-1} {\cosh (n z_2) \over \cosh(z_2)} = 2\; i^n \cosh (n z_2) \)

Note: L(n) formula derive from identity L(n) = F(2n)/F(n), and sin(2 z) = 2 sin(z) cos(z)
Note: Hyperbolics formulas derived from z₁ = i*z₂, and identities sin(z) = sinh(i*z)/i, cos(z) = cosh(i*z)