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Graphical structure of the sequence A002061(Central polygonal numbers)
Let the central part of the graphical structure in the beginning of the coordinate system (x, y) - (0,0) Each step is equal to 1.
Ie the coordinates of the first iteration
- {0,0}
second iteration (added 2 items)
- {-1, 0}
- {1, 0}
The third iteration (added 4 items)
- {-1, -1}
- {-1, 1}
- {1 -1}
- {1, 1}
etc.
If we consider each iterate and count the number of elements in the central position (x=0 and y=0) we obtain a sequence A115257
If we consider each iterate and count the number of elements in the position we obtain a sequence A006134
If we consider each iterate and count the number of elements in the position we obtain a sequence
- {0, 0, 1, 6, 27, 111, 441, 1728, 6733, 26181, 101763, ...}
have something in common with the sequence A005284
- {1, 6, 27, 111, 440, 1717, 6655, 25728, 99412, ...}
If we consider each iterate and count the number of elements in the position we obtain a sequence
- {0, 12, 212, 3152, 45488, 655328,...}
Details regarding the latest selection of items, it was observed that it suffices to consider the sum of elements at point (1,3), since the graphical construction of symmetrical.
In this case, the key point is the difference between coordinate values equal to two.
If we consider each iterate and count the number of elements in the position ; , where p-odd number, we obtain a sequence
p | x | y | sequence | factorization |
---|---|---|---|---|
1 | 3 | 1 | {0, 3, 53, 788, 11372, 163832, 2372324, 34579499, 507360379, 7489474375...} | {0, 3, 53, 2*2*197, 2*2*2843, 2*2*2*20479, 2*2*593081, 34579499, 4973*102023, 5*5*5*5*11983159...} |
3 | 5 | 3 | {0, 0, 5, 152, 3176, 57626, 977831, 16007846...} | {0, 0, 5, 2*2*2*19, 2*2*2*397, 2*28813, 977831, 2*17*470819...} |
5 | 7 | 5 | {0, 0, 0, 7, 331, 9406, 213896, 4312991...} | {0, 0, 0, 7, 331, 2*4703, 2*2*2*26737, 67*64373...} |
7 | 9 | 7 | {0, 0, 0, 0, 9, 614, 22922, 643997...} | {0, 0, 0, 0, 3*3, 2*307, 2*73*157, 83*7759...} |
- The first element of the sequence is greater than zero has the form
- {3,5,7,9...}
- The second element of the sequence is greater than zero has the form
- {53,152,331,614...}
- The third element of the sequence is greater than zero has the form
- {788,3176,9406...}
- The fourth element of the sequence is greater than zero has the form
- {11372,57626,213896...}
etc.
we can construct the following polynomials to infinity, the best of computing power.
It is interesting to consider the value of these polynomials at negative values of argument x.
As we can see there are adjacent areas.
And they form the sequence A110236.
- Number of (1,0) steps in all peakless Motzkin paths of length n (can be easily translated into RNA secondary structure terminology).
From the observed properties, we can assume that
- The Fifth Element of the sequence is greater than zero has the form
- {163832, 977831...}
- The Sixth Element of the sequence is greater than zero has the form
- {2372324, ...}
- The Seventh element of the sequence is greater than zero has the form
- {34579499, ...}
As a result, we now
The sequence of the argument x=3, I identified, because the first seven elements are of the form , where a-prime.
Please help me correct a definition of the sequence
- {,,,,,, ...}
- {3,53,788,11372,163832,2372324,34579499 ...}
- {3, 53, 2*2*197, 2*2*2843, 2*2*2*20479, 2*2*593081, 34579499...}
To add it in the encyclopedia
Unfortunately I cannot understand how to build this series up to infinity.
It should be noted another property received by polynomials. Sequence of the form
- {}
- {5,41,441,5341,68845,...}
is a sequence A115257 (Partial sums of binomial(2n,n)^2).
In other words, the sum of the coefficients of polynomials is .
The second property of polynomials
The initial idea of a generalized type of the polynomials under consideration would be:
- or
Question: what rules are formed, all the coefficients of the polynomial? Remains unresolved.
Maybe (I hope I'm not mistaken) the general formula is
where
The second coefficients of the polynomial or is the sequence A045944 (Rhombic matchstick numbers: )
The remaining coefficients of the polynomial in the OEIS are missing.
Trying to explain the graphical construction.
Graphical construction presented in the first picture, is disclosed in the two sides of the binomial theorem.