UNBOUNDED NESTED NUMBER SEQUENCES
(UNNS)
“I've often thought that there isn't any "I" at all; that we are simply the means of expression of something else; that when we think we are ourselves, we are simply the victims of a delusion.” ― Aleister Crowley, Diary of a Drug Fiend
Disclaimer:
Thus, when reading the text below, be aware that some terminology may not align with the terminology commonly adopted. The given form and meaning were expressly dictated to me. Shallow googling hasn’t brought me anything that might even slightly resemble the stuff in question that follows so far(e.g., The On-Line Encyclopedia of Integer Sequences® (OEIS®)), hence do enjoy reading it as the source of information.
To begin with, I am not a mathematician. That is what I am going to show you, which was presented to me in a dream a year ago. You may believe it or not. So consider me as an intermediary. There was not a single period in time when I tore my hair just to find something new in the realm of mathematics.
Let the numbers speak for themselves.
P.S. Keep in mind - English is not my native tongue. So, let some discrepancies be allowed.
P.S. Keep in mind - English is not my native tongue. So, let some discrepancies be allowed.
Terminology:
∟ - symbol assigned to the Nest
Nest (N) – a cipher around which a sequence is built on.
Modulus (M) – a cipher showing the order of terms in a sequence (their position in a sequence).
Index number (IN) – the first term’s value of each nested sequence by adding which
From the previous term’s value, we obtain every next term in a sequence.
Integer index (InI) – the first integer’s value of each nested sequence, by adding which to the previous integer’s value, we obtain every next integer in a sequence.
Integer (Intg) - an integer’s value
∟ - symbol assigned to the Nest
Nest (N) – a cipher around which a sequence is built on.
Modulus (M) – a cipher showing the order of terms in a sequence (their position in a sequence).
Index number (IN) – the first term’s value of each nested sequence by adding which
From the previous term’s value, we obtain every next term in a sequence.
Integer index (InI) – the first integer’s value of each nested sequence, by adding which to the previous integer’s value, we obtain every next integer in a sequence.
Integer (Intg) - an integer’s value
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Foundation:
What if one states that every existing integer possesses its own infinite set of numbers in a specific sequence that belongs strictly to it, thus forming its sequence?
What if one wants to build a sequence, where integers could be obtained a certain number of times (e.g., once per 2 terms or say once per 2 000 000 terms) in a separate sequence?
What if one can calculate the position of any integer in a given sequence using several counting methods?
Now it’s easy.
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Formula:
M∟N => SUM ((M*N)+(M/N)+(M-N)+(M+N)) (check the terminology dictionary above)
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Example 1
Let’s consider the following table.
M∟N =>((M*N)+(M/N)+(M-N)+(M+N))
1∟1 => ((1*1)+(1/1)+(1-1)+(1+1)) = (1+1+0+2) = 4 (IN - index number) 1∟1 = 4 (IN - index number)
2∟1 => (2+2+1+3) = 8 (4*2) 2∟1 = 8
3∟1 => (3+3+2+4) = 12 (4*3) 3∟1 = 12
4∟1 => (4+4+3+5) = 16 (4*4) 4∟1 = 16
5∟1 => (5+5+4+6) = 20 (4*5) 5∟1 = 20
6∟1 => (6+6+5+7) = 24 (4*6) 6∟1 = 24
7∟1 => (7+7+6+8) = 28 (4*7) 7∟1 = 28
8∟1 => (8+8+7+9) = 32 (4*8) 8∟1 = 32
9∟1 => (9+9+ 8+6+10) = 36 (4*9) 9∟1 = 36
10∟1 => (10+10+9+11) = 40 (4*10) 10∟1 = 40
…
…
…
Here we take constant number 1 as a nested element (N) => ∟1 modified by changing values of the modulus (M) M∟1, applying the formula
((M*N) + (M/N) +(M-N) + (M+N)) around which we build its sequence of numbers.
The first term in the sequence, which becomes IN (index number), represents a number by adding it to a previous one, and we obtain every next term in the sequence.
Calculation results obtained from the formula fit those obtained by adding the index number to the previous term's value accordingly.
Thus, if we want to skip building a full sequence to calculate some specific term value of a nested sequence at a certain modulus position, say there is a need to find a term value at 500 modulus position of nest 1 (500∟1), we just multiply (M) = 500 by (IN) = 4 => 500∟1 * 1∟1, resulting in 2000. Now let’s make a reverse checkup, using formula M∟N => ((M*N)+(M/N)+(M-N)+(M+N)) => 500∟1 => ((500*1)+(1/500)+(500-1)+(500+1)) = 2000.
Table 1
Let’s consider another table showing NEST (N) = ∟18.
 Judging by Example 1, it is not quite obvious that the nest denotes an integer’s position in the sequence because ∟1 represents integers only.
What about ∟18 (see Table 1), though? The table shows that the first integer (Intg 1) occurs at modulus position M18, and it is 361. Let’s check if the second position (18*2=36) of a term in the nested sequence at modulus 36∟18 remains to be an integer as well.
(M∟N) => ((m*n)+(m/n)+(m-n)+(m+n)) => 36(18*2)∟18 => ((36*18)+(36/18)+(36-18)+(36+18)) = 722
or (IN∟18)*(M36∟18 ) => 20,05555556∟18 * 36∟18 = 722
What’s even more impressive is that the first integer (Intg 1∟18) => 361 of the ∟18 sequence in Table 2 behaves just like an integer index (InI) adding which to a previous integer gives us the value of the next one.
Presumably (Intg 2∟18) + (InI∟18) => 722 + 361 = 1083 is our third integer
(Intg 3) of the nested sequence. Let’s check it up.
(M∟N) => ((M*N)+(M/N)+(M-N)+(M+N)) => 54∟18 => ((54*18)+(54/18)+(54-18)+(54+18)) =
 1083 Or (IN∟18)*(M∟18) => 20,05555556∟18 * 54∟18 = 1083
Or
InI∟18 + InI∟18 + InI∟18 = Intg3N18 => (361∟18 + 361∟18 + 361∟18 = 1083
So the first three integers of ∟18 (nest eighteen) sequence are : 18∟18 => 361, 36∟18 => 722, 54∟18 => 1083
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Table 2
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Chart 1
The table above shows 20 nested sequences at 50 modulus positions each: 1∟1 through 50∟20.
Here we can clearly observe that multiplying ∟N by M∟ with an integer value in ANY nested sequence produces an integer at the same position order in the previous nested sequence. E.G. 8∟8 (which is 81 in the table) => ∟8 * 8∟= 64 => 64 ∈ 7∟7; 9∟9 (which is 81 in the table) => ∟9 * 9∟= 81 => 81 ∈ 8∟8 (Table 3).
Table 3
Conclusions:
1. Every N (∟ - stands for NEST) of NEST (N) => ∟N constitutes the integer frequency with which it occurs (every N time for ∟N sequence) in a nested sequence, as well as an interval (each N term of N sequence is an integer) within which it occurs.
2. Every M (stands for MODULUS) => M∟N indicates the position of ANY given term in a nested (∟N) sequence.
3. Each first result of 1∟N sequence defines the Index Number of a nested sequence => (IN∟N).
4. Each first result of M∟N sequence (which becomes IN∟N) is calculated using formula 1∟N => ((1*N) + (1/N) + (1-N) + (1+N)).
((M*N) + (M/N) +(M-N) + (M+N)) The formula applies to counting ANY other result of a given nested sequence.
5. The result of IN∟N * M∟N of the same ∟N sequence produces the value of the nested sequence term at the order number M.
6. The result of ∟N * M∟ at an integer position produces an integer value at the same position order in the previous nested sequence.
UNNS as a Framework for AI Knowledge Structuring
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