Golden Satellite

Intriguing properties:

1. Fibonacci and Lucas! The sequence b_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, ...

F			G
0			2
1			1
1			3
2			4
3			7
5			11
8			18
13			29
...			...

Call this a "underground sequence s_k. Next, multiply every two consecutive red numbers

fi:	...		2		1		3		2		7		5		18		13		...
bi:		...		2		3		6		14		35		90		234		...

The sequence of curvatures in the chain is reconstructed!

2. Golden ratio \(\varphi\). The geometric proportions of the above figure contain the Golden 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 −1

The underground sequence satisfies

f_n+1  =   A·f_n − f_n-1

where the constant A alternates between the values A = 5, 1.

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