A279391 Irregular triangle read by rows in which row n lists the subparts of the successive layers of the symmetric representation of sigma(n).

1, 3, 2, 2, 7, 3, 3, 11, 1, 4, 4, 15, 5, 3, 5, 9, 9, 6, 6, 23, 5, 7, 7, 12, 12, 8, 7, 8, 1, 31, 9, 9, 35, 2, 2, 10, 10, 39, 3, 11, 5, 5, 11, 18, 18, 12, 12, 47, 13, 13, 5, 13, 21, 21, 14, 6, 6, 14, 55, 1, 15, 15, 59, 3, 7, 3, 16, 16, 63, 17, 7, 7, 17, 27, 27, 18, 9, 18, 3, 71, 10, 10, 19, 19, 30, 30

Offset

1,2

Comments

Note that the terms in the n-th row are the same as the terms in the n-th row of triangle A280851, but in some rows the terms appear in distinct order. First differs from A280851 at a(28) = T(15,3). - Omar E. Pol, Apr 24 2018

Row n in the triangle is a sequence of A250068(n) symmetric sections, each section consisting of the sizes of the subparts on that level in the symmetric representation of sigma of n - from the top down in the images below or left to right as drawn in A237593. - Hartmut F. W. Hoft, Sep 05 2021

Links

Table of n, a(n) for n=1..86.

Index entries for sequences related to sigma(n)

Example

Triangle begins (first 15 rows):
   [1];
   [3];
   [2, 2];
   [7];
   [3, 3];
   [11], [1];
   [4, 4];
   [15];
   [5, 3, 5];
   [9, 9];
   [6, 6];
   [23], [5];
   [7, 7];
   [12, 12];
   [8, 7, 8], [1];
  ...
For n = 12 we have that the 11th row of triangle A237593 is [6, 3, 1, 1, 1, 1, 3, 6] and the 12th row of the same triangle is [7, 2, 2, 1, 1, 2, 2, 7], so the diagram of the symmetric representation of sigma(12) = 28 is constructed as shown below in Figure 1:
.                          _                                    _
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                    _ _ _| |                             _ _ _| |
.                  _|    _ _|                           _|  _ _ _|
.                _|     |                             _|  _| |
.               |      _|                            |  _|  _|
.               |  _ _|                              | |_ _|
.    _ _ _ _ _ _| |    28                 _ _ _ _ _ _| |    5
.   |_ _ _ _ _ _ _|                      |_ _ _ _ _ _ _|
.                                                       23
.
.   Figure 1. The symmetric            Figure 2. After the dissection
.   representation of sigma(12)        of the symmetric representation
.   has only one part which            of sigma(12) into layers of
.   contains 28 cells, so              width 1 we can see two "subparts"
.   the 12th row of the                that contain 23 and 5 cells
.   triangle A237270 is [28].          respectively, so the 12th row of
.                                      this triangle is [23], [5].
.
For n = 15 we have that the 14th row of triangle A237593 is [8, 3, 1, 2, 2, 1, 3, 8] and the 15th row of the same triangle is [8, 3, 2, 1, 1, 1, 1, 2, 3, 8], so the diagram of the symmetric representation of sigma(15) = 24 is constructed as shown below in Figure 3:
.                                _                                  _
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                          _ _ _|_|                           _ _ _|_|
.                      _ _| |      8                      _ _| |      8
.                     |    _|                            |  _ _|
.                    _|  _|                             _| |_|
.                   |_ _|  8                           |_ _|  1
.                   |                                  |    7
.    _ _ _ _ _ _ _ _|                   _ _ _ _ _ _ _ _|
.   |_ _ _ _ _ _ _ _|                  |_ _ _ _ _ _ _ _|
.                    8                                  8
.
.   Figure 3. The symmetric            Figure 4. After the dissection
.   representation of sigma(15)        of the symmetric representation
.   has three parts of size 8          of sigma(15) into layers of
.   because every part contains        width 1 we can see four "subparts".
.   8 cells, so the 15th row of        The first layer has three subparts:
.   triangle A237270 is [8, 8, 8].     8, 7, 8. The second layer has
.                                      only one subpart of size 1, so
.                                      the 15th row of this triangle is
.                                      [8, 7, 8], [1].
.
The smallest even number with 3 levels is 60; its row of subparts is: [119], [37], [6, 6]. The smallest odd number with 3 levels is 315; its row of subparts is:  [158, 207, 158], [11, 26, 5, 9, 5, 26, 11], [4, 4]. - Hartmut F. W. Hoft, Sep 05 2021

Mathematica

(* support functions are defined in aA237593 and A262045 *)
subP[level_] := Module[{s=Map[Apply[Plus, #]&, Select[level, First[#]!=0&]]}, If[OddQ[Length[s]], s[[(Length[s]+1)/2]]-=1]; s]
a279391[n_] := Module[{widL=a262045[n], lenL=a237593[n], srs, subs}, srs=Transpose[Map[PadRight[If[widL[[#]]>0, Table[1, widL[[#]]], {0}], Max[widL]]&, Range[Length[lenL]]]]; subs=Map[SplitBy[lenL srs[[#]], #!=0&]&, Range[Max[widL]]]; Flatten[Map[subP, subs]]]
Flatten[Map[a279391, Range[38]]] (* Hartmut F. W. Hoft, Sep 05 2021 *)

Crossrefs

The length of row n equals A001227(n).

If n is odd the length of row n equals A000005(n).

Row sums give A000203.

For the definition of "subparts" see A279387.

For the triangle of sums of subparts see A279388.

Cf. A001227, A005279, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239657, A244050, A245092, A250068, A250070, A261699, A262626, A280851, A296508.

Sequence in context: A235773 A089327 A280851 * A237270 A091264 A021760

Adjacent sequences: A279388 A279389 A279390 * A279392 A279393 A279394

Keyword

nonn,tabf

Author

Omar E. Pol, Dec 12 2016

Status

approved