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I am sorry, but I don't get it again. If that sequence would reach a very large number n, and continue consistently on for as long as we can follow, according to the same rule, as an output of a random number generator, I wouldn't call it perfect, but extremely poorly designed random number generator. In fact I don't know what could be worse than that. Maybe to produce a dice for a sole reason of using it for random number generation, and whenever you throw it, you get the same number? Or a random number generator that produces such output: {1,2,3,4,5, ...}? If I am not missing something here, that would be the same class of failure. Right?

I mean, if you construct something that is not supposed to output something that can be described by such a simple rule, and you still get it, then you didn't do a good job.

OK, I got it you are right, there is no reason why should we expect that random sequences cannot turn out to be easily described by simple rules.

Especially if we believe that nature is regular in its essence.

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So, the question is, why are random sequences in reality, in nature, not like that, describable by simple rules? With a perfectly balanced dice, getting every time the same number would be in conflict with the law of large numbers. I think I am at the end of my understanding am I confronted here with some oddness or not. My intuition tells me that random number sequence cannot be generated by a simple rule, because, it is not generated by any rule, by definition, and that actually tells me my logic too. Maybe you were seeking to much for some mystery here, when there is none, studiot?

Although, that is also some kind of rule (no rule).

And it's a simple one.

And if you draw a random line in a coordinate system, with a free hand, that is, without a ruler and compass, I bet it would be asymmetric, for the same reason.

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On ‎5‎/‎18‎/‎2019 at 12:39 PM, studiot said:

Consider the sequence

{1, 2, 4, 7, 11, 16 ...}﻿﻿

This can be generated in a variety of ways,

Actually, the sequence that you mentioned can be described only in these two ways, using a recurrence relation $x_n=x_{n-1}+n , \forall{n>0}$, which is an example of a first order linear difference equation, with initial condition $x_0=1$, and by its closed-form solution $x_n=1+\frac{n(n+1)}{2}$. That this formula is a solution of that difference equation can be shown like this:

$x_1=x_0+1$
$x_2=x_1+2=x_0+1+2$
$x_3=x_2+3=x_0+1+2+3$
...
$x_n=x_0+1+2+3+...+n=x_0+\sum_{k=1}^{n} k=1+\frac{n(n+1)}{2}$

$x_{n-1}=1+\frac{(n-1)n}{2}$

$x_n-x_{n-1}=1-1+\frac{n}{2} (n+1-(n-1))=\frac{n}{2}(1+1)=n$

The fact that this sequence cannot be output of a true random generator is a fine example of regularity of nature, which can be expressed by a rule that when certain outcome is not enforced (by some rule, algorithm or physical constraint), it does not happen, because there is a multitude of other, equally probable possibilities. So, finite sequence {1, 2, 4, 7, 11, 16}﻿﻿ can be easily a result of random choosing of 6 numbers from a certain range of numbers, for example from the first 16 natural numbers, but the regular infinite sequence {1, 2, 4, 7, 11, 16 ...}﻿﻿ cannot be a result of random choosing among all natural numbers.

On ‎5‎/‎16‎/‎2019 at 1:21 PM, studiot said:

The whole beauty of the English language is that it has so many words with similar meanings and even substantial overlap of meaning.
This is because there is so many subtle variations of meaning available.

While the "random" sequence was mildly interesting contribution to the discussion, this was a very pedestrian observation. In fact this is even worse than that, this could have been pedestrian if you mentioned instead the beauty of any natural language, which, for your information, share the same trait.

Edited by Hrvoje1

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19 hours ago, Hrvoje1 said:

when﻿ certain ﻿outcome is not enforced (by som﻿e rule, algorithm or physical constraint), it does not happe﻿n﻿﻿,﻿

I mean, when certain outcome is not deterministically caused, it happens according to its probability, which is in this case zero. At least in an experiment that is repeated an infinite number of times, such as here. Choosing one infinite sequence among an infinite number of such sequences, means choosing one number among an infinite number of natural numbers, infinitely many times, to form that infinite sequence. Right?

20 hours ago, Hrvoje1 said:

th﻿e﻿re is ﻿﻿a﻿ mult﻿itude of﻿ ﻿other, equal﻿ly﻿ ﻿probable﻿ p﻿oss﻿ibil﻿iti﻿﻿﻿es﻿

Infinite random sequences are all equally possible outputs of true random number generator, while deterministic infinite sequences are all equally impossible.

The only problem with randomness is that the Bourbaki school considered the statement "let us consider a random sequence" an abuse of language.

Edited by Hrvoje1

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