Gentle dynamics

 

It is well known that dynamic systems are typically unstoppable, so, as Poincaré has already shown, there is some mysterious obstacle in the way of any long-term forecast.

A dynamic system can consist of many, but it can even consist of a variable, in which case we can speak of a simple sequence. But what makes a system or a sequence dynamic. Basically, because it is a function of time, namely discrete, it is a function of time consisting of some unit of time. This will really be the function sequence.

But dynamic systems have another important feature: the new values ​​depend on the “past”, that is, not only on the value of the variable, but on the values ​​taken in previous places in the function - in the case of a sequence, on the preceding members.

One of the best known dynamic sequence is Fibonacci numbers. This is not really a family of one, but an infinite number of sequence with very similar properties. These are truly dizzyingly exciting, thoroughly researched sequence, and by the way not so extreme, yet even in the gentlest version, today’s Excel around the 50th member “throws in the towel”.

Meanwhile, it must be acknowledged that it is difficult to imagine a more gentle than the Fibonacci formula. Anyone trying to compose some kind of dynamic sequence faces a similar difficulty: values ​​grow at a dizzying pace over time, even if some cyclicality develops in the meantime. Therefore, it was a pleasant surprise to find an absolutely gentle dynamic sequence that also works with the square of values.

Here's the sequence:

 

1, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 16, 15, 16, 17, 16, 17, 18, 17, 18, 19, 18, 19, 20, 19, 20, 21, 20, 21, 22, 21, 22, …

 

No doubt, it is really gentle and in a very nice dance move, a real szirtaki! What formula could he have born?

Here is the formula for the sequence:

 

a(n) = a(n-2)^2 + a(n-1)^2 – 2*a(n-2)*a(n-1) – a(n-2) – a(n-1) +n + 1

 

You can try with other starting values and you will immediately experience the aforementioned difficulties of prognosis.

But why this strange three-step cycle?

 

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