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%I #184 Feb 14 2022 21:36:07
%S 1,3,2,2,7,3,3,12,4,4,15,5,3,5,9,9,6,6,28,7,7,12,12,8,8,8,31,9,9,39,
%T 10,10,42,11,5,5,11,18,18,12,12,60,13,5,13,21,21,14,6,6,14,56,15,15,
%U 72,16,16,63,17,7,7,17,27,27,18,12,18,91,19,19,30,30,20,8,8,20,90
%N Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(n).
%C T(n,k) is the number of cells in the k-th region of the n-th set of regions in a diagram of the symmetry of sigma(n), see example.
%C Row n is a palindromic composition of sigma(n).
%C Row sums give A000203.
%C Row n has length A237271(n).
%C In the row 2n-1 of triangle both the first term and the last term are equal to n.
%C If n is an odd prime then row n is [m, m], where m = (1 + n)/2.
%C The connection with A196020 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A239660 --> this sequence.
%C For the boundary segments in an octant see A237591.
%C For the boundary segments in a quadrant see A237593.
%C For the boundary segments in the spiral see also A239660.
%C For the parts in every quadrant of the spiral see A239931, A239932, A239933, A239934.
%C We can find the spiral on the terraces of the stepped pyramid described in A244050. - _Omar E. Pol_, Dec 07 2016
%C T(n,k) is also the area of the k-th terrace, from left to right, at the n-th level, starting from the top, of the stepped pyramid described in A245092 (see Links section). - _Omar E. Pol_, Aug 14 2018
%H Robert Price, <a href="/A237270/b237270.txt">Table of n, a(n) for n = 1..15542</a> (rows n = 1..5000, flattened)
%H Hartmut F. W. Hoft, <a href="/A237270/a237270_1.pdf">Sample visual documentation for Mathematica code</a>
%H Michel Marcus, <a href="/A244145/a244145_3.pdf">A colored version of the symmetric representation of sigma(n), multipage, n = 1..85</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr01.jpg">An infinite stepped pyramid (A237593, A237270, A262626)</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (16 levels)</a>
%H Omar E. Pol, <a href="/A237270/a237270.jpg">Perspective view of the stepped pyramid into four quadrants (11 levels)</a>. This is formed by combing four copies of the pyramid back-to-back (cf. A244050).
%H N. J. A. Sloane, <a href="/A237270/a237270_2.pdf">Another drawing of the spiral</a>
%H N. J. A. Sloane, <a href="/A237270/a237270_3.pdf">Spiral showing only the outer boundary</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e Illustration of the first 27 terms as regions (or parts) of a spiral constructed with the first 15.5 rows of A239660:
%e .
%e . _ _ _ _ _ _ _ _
%e . | _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
%e . | | |_ _ _ _ _ _ _|
%e . 12 _| | |
%e . |_ _| _ _ _ _ _ _ |_ _
%e . 12 _ _| | _ _ _ _ _|_ _ _ _ _ 5 |_
%e . _ _ _| | 9 _| | |_ _ _ _ _| |
%e . | _ _ _| 9 _|_ _| |_ _ 3 |_ _ _ 7
%e . | | _ _| | _ _ _ _ |_ | | |
%e . | | | _ _| 12 _| _ _ _|_ _ _ 3 |_|_ _ 5 | |
%e . | | | | _| | |_ _ _| | | | |
%e . | | | | | _ _| |_ _ 3 | | | |
%e . | | | | | | 3 _ _ | | | | | |
%e . | | | | | | | _|_ 1 | | | | | |
%e . _|_| _|_| _|_| _|_| |_| _|_| _|_| _|_| _
%e . | | | | | | | | | | | | | | | |
%e . | | | | | | |_|_ _ _| | | | | | | |
%e . | | | | | | 2 |_ _|_ _| _| | | | | | |
%e . | | | | |_|_ 2 |_ _ _|7 _ _| | | | | |
%e . | | | | 4 |_ _| _ _| | | | |
%e . | | |_|_ _ |_ _ _ _ | _| _ _ _| | | |
%e . | | 6 |_ |_ _ _ _|_ _ _ _| | 15 _| _ _| | |
%e . |_|_ _ _ |_ 4 |_ _ _ _ _| _| | _ _ _| |
%e . 8 | |_ _ | | _| | _ _ _|
%e . |_ | |_ _ _ _ _ _ | _ _|28 _| |
%e . |_ |_ |_ _ _ _ _ _|_ _ _ _ _ _| | _| _|
%e . 8 |_ _| 6 |_ _ _ _ _ _ _| _ _| _|
%e . | | _ _| 31
%e . |_ _ _ _ _ _ _ _ | |
%e . |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
%e . 8 |_ _ _ _ _ _ _ _ _|
%e .
%e .
%e [For two other drawings of the spiral see the links. - _N. J. A. Sloane_, Nov 16 2020]
%e If the sequence does not contain negative terms then its terms can be represented in a quadrant. For the construction of the diagram we use the symmetric Dyck paths of A237593 as shown below:
%e ---------------------------------------------------------------
%e Triangle Diagram of the symmetry of sigma (n = 1..24)
%e ---------------------------------------------------------------
%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e 1; |_| | | | | | | | | | | | | | | | | | | | | | | |
%e 3; |_ _|_| | | | | | | | | | | | | | | | | | | | | |
%e 2, 2; |_ _| _|_| | | | | | | | | | | | | | | | | | | |
%e 7; |_ _ _| _|_| | | | | | | | | | | | | | | | | |
%e 3, 3; |_ _ _| _| _ _|_| | | | | | | | | | | | | | | |
%e 12; |_ _ _ _| _| | _ _|_| | | | | | | | | | | | | |
%e 4, 4; |_ _ _ _| |_ _|_| _ _|_| | | | | | | | | | | |
%e 15; |_ _ _ _ _| _| | _ _ _|_| | | | | | | | | |
%e 5, 3, 5; |_ _ _ _ _| | _|_| | _ _ _|_| | | | | | | |
%e 9, 9; |_ _ _ _ _ _| _ _| _| | _ _ _|_| | | | | |
%e 6, 6; |_ _ _ _ _ _| | _| _| _| | _ _ _ _|_| | | |
%e 28; |_ _ _ _ _ _ _| |_ _| _| _ _| | | _ _ _ _|_| |
%e 7, 7; |_ _ _ _ _ _ _| | _ _| _| _| | | _ _ _ _|
%e 12, 12; |_ _ _ _ _ _ _ _| | | | _|_| |* * * *
%e 8, 8, 8; |_ _ _ _ _ _ _ _| | _ _| _ _|_| |* * * *
%e 31; |_ _ _ _ _ _ _ _ _| | _ _| _| _ _|* * * *
%e 9, 9; |_ _ _ _ _ _ _ _ _| | |_ _ _| _|* * * * * *
%e 39; |_ _ _ _ _ _ _ _ _ _| | _ _| _|* * * * * * *
%e 10, 10; |_ _ _ _ _ _ _ _ _ _| | | |* * * * * * * *
%e 42; |_ _ _ _ _ _ _ _ _ _ _| | _ _ _|* * * * * * * *
%e 11, 5, 5, 11; |_ _ _ _ _ _ _ _ _ _ _| | |* * * * * * * * * * *
%e 18, 18; |_ _ _ _ _ _ _ _ _ _ _ _| |* * * * * * * * * * *
%e 12, 12; |_ _ _ _ _ _ _ _ _ _ _ _| |* * * * * * * * * * *
%e 60; |_ _ _ _ _ _ _ _ _ _ _ _ _|* * * * * * * * * * *
%e ...
%e The total number of cells in the first n set of symmetric regions of the diagram equals A024916(n), the sum of all divisors of all positive integers <= n, hence the total number of cells in the n-th set of symmetric regions of the diagram equals sigma(n) = A000203(n).
%e For n = 9 the 9th row of A237593 is [5, 2, 2, 2, 2, 5] and the 8th row of A237593 is [5, 2, 1, 1, 2, 5] therefore between both symmetric Dyck paths there are three regions (or parts) of sizes [5, 3, 5], so row 9 is [5, 3, 5].
%e The sum of divisors of 9 is 1 + 3 + 9 = A000203(9) = 13. On the other hand the sum of the parts of the symmetric representation of sigma(9) is 5 + 3 + 5 = 13, equaling the sum of divisors of 9.
%e For n = 24 the 24th row of A237593 is [13, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 13] and the 23rd row of A237593 is [12, 5, 2, 2, 1, 1, 1, 1, 2, 2, 5, 12] therefore between both symmetric Dyck paths there are only one region (or part) of size 60, so row 24 is 60.
%e The sum of divisors of 24 is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = A000203(24) = 60. On the other hand the sum of the parts of the symmetric representation of sigma(24) is 60, equaling the sum of divisors of 24.
%e Note that the number of *'s in the diagram is 24^2 - A024916(24) = 576 - 491 = A004125(24) = 85.
%e From _Omar E. Pol_, Nov 22 2020: (Start)
%e Also consider the infinite double-staircases diagram defined in A335616 (see the theorem).
%e For n = 15 the diagram with first 15 levels looks like this:
%e .
%e Level "Double-staircases" diagram
%e . _
%e 1 _|1|_
%e 2 _|1 _ 1|_
%e 3 _|1 |1| 1|_
%e 4 _|1 _| |_ 1|_
%e 5 _|1 |1 _ 1| 1|_
%e 6 _|1 _| |1| |_ 1|_
%e 7 _|1 |1 | | 1| 1|_
%e 8 _|1 _| _| |_ |_ 1|_
%e 9 _|1 |1 |1 _ 1| 1| 1|_
%e 10 _|1 _| | |1| | |_ 1|_
%e 11 _|1 |1 _| | | |_ 1| 1|_
%e 12 _|1 _| |1 | | 1| |_ 1|_
%e 13 _|1 |1 | _| |_ | 1| 1|_
%e 14 _|1 _| _| |1 _ 1| |_ |_ 1|_
%e 15 |1 |1 |1 | |1| | 1| 1| 1|
%e .
%e Starting from A196020 and after the algorithm described in A280850 and A296508 applied to the above diagram we have a new diagram as shown below:
%e .
%e Level "Ziggurat" diagram
%e . _
%e 6 |1|
%e 7 _ | | _
%e 8 _|1| _| |_ |1|_
%e 9 _|1 | |1 1| | 1|_
%e 10 _|1 | | | | 1|_
%e 11 _|1 | _| |_ | 1|_
%e 12 _|1 | |1 1| | 1|_
%e 13 _|1 | | | | 1|_
%e 14 _|1 | _| _ |_ | 1|_
%e 15 |1 | |1 |1| 1| | 1|
%e .
%e The 15th row
%e of A249351 : [1,1,1,1,1,1,1,1,0,0,0,1,1,1,2,1,1,1,0,0,0,1,1,1,1,1,1,1,1]
%e The 15th row
%e of triangle: [ 8, 8, 8 ]
%e The 15th row
%e of A296508: [ 8, 7, 1, 0, 8 ]
%e The 15th row
%e of A280851 [ 8, 7, 1, 8 ]
%e .
%e More generally, for n >= 1, it appears there is the same correspondence between the original diagram of the symmetric representation of sigma(n) and the "Ziggurat" diagram of n.
%e For the definition of subparts see A239387 and also A296508, A280851. (End)
%t T[n_,k_] := Ceiling[(n + 1)/k - (k + 1)/2] (* from A235791 *)
%t path[n_] := Module[{c = Floor[(Sqrt[8n + 1] - 1)/2], h, r, d, rd, k, p = {{0, n}}}, h = Map[T[n, #] - T[n, # + 1] &, Range[c]]; r = Join[h, Reverse[h]]; d = Flatten[Table[{{1, 0}, {0, -1}}, {c}], 1];
%t rd = Transpose[{r, d}]; For[k = 1, k <= 2c, k++, p = Join[p, Map[Last[p] + rd[[k, 2]] * # &, Range[rd[[k, 1]]]]]]; p]
%t segments[n_] := SplitBy[Map[Min, Drop[Drop[path[n], 1], -1] - path[n - 1]], # == 0 &]
%t a237270[n_] := Select[Map[Apply[Plus, #] &, segments[n]], # != 0 &]
%t Flatten[Map[a237270, Range[40]]] (* data *)
%t (* _Hartmut F. W. Hoft_, Jun 23 2014 *)
%Y Cf. A000203, A004125, A023196, A024916, A153485, A196020, A221529, A231347, A235791, A235796, A236104, A236112, A236540, A237046, A237048, A237271, A237590, A237591, A237593, A239050, A239660, A239663, A239665, A239931, A239932, A239933, A239934, A240020, A240062, A244050, A245092, A249351, A262626, A280850, A280851, A296508, A335616, A340035.
%K nonn,tabf,look
%O 1,2
%A _Omar E. Pol_, Feb 19 2014
%E Drawing of the spiral extended by _Omar E. Pol_, Nov 22 2020