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Triangle read by rows in which row n lists the widths of the symmetric representation of sigma(n).
63

%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