This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A279391 #96 Sep 05 2021 15:19:32
%S A279391 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,
%T A279391 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,
%U A279391 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
%N A279391 Irregular triangle read by rows in which row n lists the subparts of the successive layers of the symmetric representation of sigma(n).
%C A279391 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
%C A279391 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
%H A279391 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A279391 Triangle begins (first 15 rows):
%e A279391 [1];
%e A279391 [3];
%e A279391 [2, 2];
%e A279391 [7];
%e A279391 [3, 3];
%e A279391 [11], [1];
%e A279391 [4, 4];
%e A279391 [15];
%e A279391 [5, 3, 5];
%e A279391 [9, 9];
%e A279391 [6, 6];
%e A279391 [23], [5];
%e A279391 [7, 7];
%e A279391 [12, 12];
%e A279391 [8, 7, 8], [1];
%e A279391 ...
%e A279391 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:
%e A279391 . _ _
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%e A279391 . _| _ _| _| _ _ _|
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%e A279391 . | _| | _| _|
%e A279391 . | _ _| | |_ _|
%e A279391 . _ _ _ _ _ _| | 28 _ _ _ _ _ _| | 5
%e A279391 . |_ _ _ _ _ _ _| |_ _ _ _ _ _ _|
%e A279391 . 23
%e A279391 .
%e A279391 . Figure 1. The symmetric Figure 2. After the dissection
%e A279391 . representation of sigma(12) of the symmetric representation
%e A279391 . has only one part which of sigma(12) into layers of
%e A279391 . contains 28 cells, so width 1 we can see two "subparts"
%e A279391 . the 12th row of the that contain 23 and 5 cells
%e A279391 . triangle A237270 is [28]. respectively, so the 12th row of
%e A279391 . this triangle is [23], [5].
%e A279391 .
%e A279391 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:
%e A279391 . _ _
%e A279391 . | | | |
%e A279391 . | | | |
%e A279391 . | | | |
%e A279391 . | | | |
%e A279391 . | | | |
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%e A279391 . | | | |
%e A279391 . _ _ _|_| _ _ _|_|
%e A279391 . _ _| | 8 _ _| | 8
%e A279391 . | _| | _ _|
%e A279391 . _| _| _| |_|
%e A279391 . |_ _| 8 |_ _| 1
%e A279391 . | | 7
%e A279391 . _ _ _ _ _ _ _ _| _ _ _ _ _ _ _ _|
%e A279391 . |_ _ _ _ _ _ _ _| |_ _ _ _ _ _ _ _|
%e A279391 . 8 8
%e A279391 .
%e A279391 . Figure 3. The symmetric Figure 4. After the dissection
%e A279391 . representation of sigma(15) of the symmetric representation
%e A279391 . has three parts of size 8 of sigma(15) into layers of
%e A279391 . because every part contains width 1 we can see four "subparts".
%e A279391 . 8 cells, so the 15th row of The first layer has three subparts:
%e A279391 . triangle A237270 is [8, 8, 8]. 8, 7, 8. The second layer has
%e A279391 . only one subpart of size 1, so
%e A279391 . the 15th row of this triangle is
%e A279391 . [8, 7, 8], [1].
%e A279391 .
%e A279391 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
%t A279391 (* support functions are defined in aA237593 and A262045 *)
%t A279391 subP[level_] := Module[{s=Map[Apply[Plus, #]&, Select[level, First[#]!=0&]]}, If[OddQ[Length[s]], s[[(Length[s]+1)/2]]-=1]; s]
%t A279391 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]]]
%t A279391 Flatten[Map[a279391, Range[38]]] (* _Hartmut F. W. Hoft_, Sep 05 2021 *)
%Y A279391 The length of row n equals A001227(n).
%Y A279391 If n is odd the length of row n equals A000005(n).
%Y A279391 Row sums give A000203.
%Y A279391 For the definition of "subparts" see A279387.
%Y A279391 For the triangle of sums of subparts see A279388.
%Y A279391 Cf. A001227, A005279, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239657, A244050, A245092, A250068, A250070, A261699, A262626, A280851, A296508.
%K A279391 nonn,tabf
%O A279391 1,2
%A A279391 _Omar E. Pol_, Dec 12 2016