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%I #81 Aug 02 2023 14:33:32
%S 1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,1,1,1,1,1,1,1,2,1,1,1,
%T 1,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,0,0,1,1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1
%N Triangle read by rows in which row n lists the widths of the symmetric representation of sigma(n).
%C Here T(n,k) is defined to be the "k-th width" of the symmetric representation of sigma(n), with n>=1 and 1<=k<=2n-1. Explanation: consider the diagram of the symmetric representation of sigma(n) described in A236104, A237593 and other related sequences. Imagine that the diagram for sigma(n) contains 2n-1 equidistant segments which are parallel to the main diagonal [(0,0),(n,n)] of the quadrant. The segments are located on the diagonal of the cells. The distance between two parallel segment is equal to sqrt(2)/2. T(n,k) is the length of the k-th segment divided by sqrt(2). Note that the triangle contains nonnegative terms because for some n the value of some widths is equal to zero. For an illustration of some widths see _Hartmut F. W. Hoft_'s contribution in the Links section of A237270.
%C Row n has length 2*n-1.
%C Row sums give A000203.
%C If n is a power of 2 then all terms of row n are 1's.
%C If n is an even perfect number then all terms of row n are 1's except the middle term which is 2.
%C If n is an odd prime then row n lists (n+1)/2 1's, n-2 zeros, (n+1)/2 1's.
%C The number of blocks of positive terms in row n gives A237271(n).
%C The sum of the k-th block of positive terms in row n gives A237270(n,k).
%C It appears that the middle diagonal is also A067742 (which was conjectured by _Michel Marcus_ in the entry A237593 and checked with two Mathematica functions up to n = 100000 by _Hartmut F. W. Hoft_).
%C It appears that the trapezoidal numbers (A165513) are also the numbers k > 1 with the property that some of the noncentral widths of the symmetric representation of sigma(k) are not equal to 1. - _Omar E. Pol_, Mar 04 2023
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e Triangle begins:
%e 1;
%e 1,1,1;
%e 1,1,0,1,1;
%e 1,1,1,1,1,1,1;
%e 1,1,1,0,0,0,1,1,1;
%e 1,1,1,1,1,2,1,1,1,1,1;
%e 1,1,1,1,0,0,0,0,0,1,1,1,1;
%e 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
%e 1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,1,1;
%e 1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1;
%e 1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1;
%e 1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1;
%e ...
%e ---------------------------------------------------------------------------
%e . Written as an isosceles triangle Diagram of
%e . the sequence begins: the symmetry of sigma
%e ---------------------------------------------------------------------------
%e . _ _ _ _ _ _ _ _ _ _ _ _
%e . 1; |_| | | | | | | | | | | |
%e . 1,1,1; |_ _|_| | | | | | | | | |
%e . 1,1,0,1,1; |_ _| _|_| | | | | | | |
%e . 1,1,1,1,1,1,1; |_ _ _| _|_| | | | | |
%e . 1,1,1,0,0,0,1,1,1; |_ _ _| _| _ _|_| | | |
%e . 1,1,1,1,1,2,1,1,1,1,1; |_ _ _ _| _| | _ _|_| |
%e . 1,1,1,1,0,0,0,0,0,1,1,1,1; |_ _ _ _| |_ _|_| _ _|
%e . 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1; |_ _ _ _ _| _| |
%e . 1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,1,1; |_ _ _ _ _| | _|
%e . 1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1; |_ _ _ _ _ _| _ _|
%e . 1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1; |_ _ _ _ _ _| |
%e .1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1; |_ _ _ _ _ _ _|
%e ...
%e From _Omar E. Pol_, Nov 22 2020: (Start)
%e Also consider the infinite double-staircases diagram defined in A335616.
%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 this seq: [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 A237270: [ 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 The number of horizontal steps (or 1's) in the successive columns of the above diagram gives the 15th row of this triangle.
%e For more information about the parts of the symmetric representation of sigma(n) see A237270. For more information about the subparts see A239387, A296508, A280851.
%e More generally, it appears there is the same correspondence between the original diagram of the symmetric representation of sigma(n) and the "Ziggurat" diagram of n. (End)
%t (* function segments are defined in A237270 *)
%t a249351[n_] := Flatten[Map[segments, Range[n]]]
%t a249351[10] (* _Hartmut F. W. Hoft_, Jul 20 2022 *)
%Y Cf. A000203, A003056, A067742, A071562, A165513, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A238443, A239660, A239932-A239934, A240542, A241008, A241010, A245092, A245685, A246955, A246956, A247687, A249223, A250068, A250070, A250071, A262626, A280850, A280851, A296508, A235616, A347186.
%K nonn,tabf
%O 1,31
%A _Omar E. Pol_, Oct 26 2014