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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Aristarchus sequences

While writing my book The Numbers, I came up with dozens of more or less interesting sequences. I wrote some for myself, not others, but in each case I had the intention of returning to them “almost once”.

So in the days I took out one and discovered some very interesting qualities in him and - to my delight - all the more mystery.

Since it’s a whole family of sequences, I felt it would be nice to give them a name. So I started calling myself the Aristarchus sequences. He who accepts this designation accepts who does not. In any case, Aristarchus deserves our respect.

But let's move on!

The basic sequences is formed according to the following rule:

a(1) = 1, if n> 1, a(n) = a(n-1)/gcd(a (n-1), n) if gcd(a(n-1), n) > 1, if gcd(a(n-1), n) = 1, then a(n) = a(n-1) + n.

In words, if the largest common divisor of the previous member and n is greater than 1) so is a real divisor), then we divide the previous member by that, and this becomes the new member, but if the previous member and n are relative prime, the new member is the sum of the previous member and n.

There are no special “screws” in the rule itself, we can wait with interest for the result.

Well, that might even seem disappointing, because this sequences looks like this:

1, 3, 1, 5, 1, 7, 1, 9, 1, ….

That is, it is fairly unanimous and predictable.

If that were all, we might even forget the sequences.

But let’s see what happens if the first member is not 1, but - for the sake of order - 2!

A little surprise. The sequences now looks like this:

2, 1, 4, 1, 6, 1, 8, 1, …

Exemplary order and discipline, almost beautiful! While every sequences would be like this, we would have no problem with them.

But the mathematician is curious, let’s see how this sequences looks like for other novice members when we talk about it so much!

Let's see what the sequences is like if the first member is 3!

Well, you should hold on now! The sequences looks like this:

3, 5, 8, 2, 7, 13, 20, 5, 14, 7, 18, 3, 16, 8, 23, 39, 56, 28, 47, 67, 88, 4, 27, 9, 34, 17, 44, 11, 40, 4, 35, 67, 100, 50, 10, 5, 42, 21, 7, 47, 88, 44, 87, 131, 176, 88, 135, 45, 94, 47, 98, 49, 102, 17, 72, 9, 3, 61, 120, 2, 63, 125, 188, 47, 112, 56, 123, 191, 260, 26, 97, 169, 242, 121, 196, 49, 7, 85, 164, 41, 122, 61, 144, 12, 97, 183, 61, 149, 238, 119, 17, 109, 202, 101, 196, 49, 146, 73, 172, 43, 144, 24, 127, 231, 11, 117, 224, 56, 165, 3, …

Undoubtedly, this looks staggering after the first two cases. You could say a real random sequence. Indeed, what connections can be discovered here? (Warning: this is just a provocative question.)

And now may come the real surprise! Right here, where I left off, this bizarre unruly sequence returns to the well-known chiseled bed and continues like this:

1, 113, 1, 115, 1, 117, 1, 119, 1, …

The case is, to put it mildly, interesting, and once discovered, an honest numerator does not leave it at that, but examines it with all natural numbers. (Not big so, this is the smallest infinite set.)

What a denial, this is also an interesting sequence when the sequences returns to normal.

Here are the modest first results (here I will only give the results that form a continuous line):

1, 2, 111, 7, 5, 3, 25, 22, 25, 111, 111, 4, 7, 5, 5, 6, 22, …

Why does 111 occur so many times, why does it occur twice in a row (if the sequences starts with 10 or 11)?

Dozens of questions, dozens of mysteries. We will certainly never get an answer to some of them, yes to others. And I'm looking forward to them.

Finally, another important additive to liven up life in the world of the Aristarchus sequences. Well, a small change in the basic formula dramatically changes its character, leaving no trace of this old monotonous bed. And this change is that if gcd is 1, then the new term is the sum of the previous and n, minus 1. Then the sequence looks like this:

1, 2, 4, 1, 5, 10, 16, 2, 10, 1, 11, 22, 34, 17, 31, 46, 62, 31, 49, 68, 88, 4, 26, 13, 37, 62, 88, 22, 50, 5, 35, 66, 2, 1, 35, 70, 106, 53, 91, 130, 170, 85, 127, 170, 34, 17, 63, 21, 3, 52, 102, 51, 103, 156, 210, 15, 5, 62, 120, 2, 62, 1, 63, 126, 190, 95, 161, 228, 76, 38, 108, 3, 75, 148, 222, 111, 187, 264, 342, 171, 19, …

We see here that the appearance of 1 does not lead the sequences anywhere, boredom is ruled out.

That is now the first news of the Aristarchus sequences.

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