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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