## Golden Satellite

Intriguing properties:

1. Fibonacci and Lucas!The sequenceb_{i}of curvatures:

...14, 6, 3, 2, 2, 3, 6, 14, 30, 90, ... may be obtained from Lucas and Fibonacci sequences. Below, the first column represents Fibonacci numbers (F), the second -- Lucas numbers (L). Consider every other number in each sequence, as indicated by bold characters. They form a zig-zag sequence: 2, 1, 3, 2, 7, 8, 18, ...

Call this a "underground sequence

FG0 211 1 324 3 7511 8 181329 ... ... s. Next, multiply every two consecutive red numbers_{k}The sequence of curvatures in the chain is reconstructed!

f _{i}:... 2 1 3 2 7 5 18 13 ... b_{i}:... 2 3 6 14 35 90 234 ...

2. Golden ratio \(\varphi\).The geometric proportions of the above figure contain theGolden ratio. To see it, press any key. It will show the golden rectangle and the golden proportion (toggle with any key). The suggested identity: $$ \frac{1}{2} + \frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \frac{1}{14}\ldots = \frac{1+\sqrt{5}}{2} \equiv \varphi $$

3. Recurrence.The sequence of curvatures follows a nonhomogeneous recurrence:b_{n+1}= 3b_{n}−b_{n-1}−1The underground sequence satisfies

f _{n+1}=A·f_{n}− f_{n-1}where the constant

Aalternates between the valuesA= 5, 1.

Thanks to Christian Rose who spotted some numerical typos in the original page