A240020 -id:A240020 - cp's OEIS frontend

A237270 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(n).

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, 10, 10, 42, 11, 5, 5, 11, 18, 18, 12, 12, 60, 13, 5, 13, 21, 21, 14, 6, 6, 14, 56, 15, 15, 72, 16, 16, 63, 17, 7, 7, 17, 27, 27, 18, 12, 18, 91, 19, 19, 30, 30, 20, 8, 8, 20, 90

Offset: 1

Omar E. Pol, Feb 19 2014

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.
Row n is a palindromic composition of sigma(n).
Row sums give A000203.
Row n has length A237271(n).
In the row 2n-1 of triangle both the first term and the last term are equal to n.
If n is an odd prime then row n is [m, m], where m = (1 + n)/2.
The connection with A196020 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A239660 --> this sequence.
For the boundary segments in an octant see A237591.
For the boundary segments in a quadrant see A237593.
For the boundary segments in the spiral see also A239660.
For the parts in every quadrant of the spiral see A239931, A239932, A239933, A239934.
We can find the spiral on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016
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

			Illustration of the first 27 terms as regions (or parts) of a spiral constructed with the first 15.5 rows of A239660:
.
.                  _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.             12 _| |                           |
.               |_ _|  _ _ _ _ _ _              |_ _
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_
.      _ _ _| |    9 _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |      _ _| |      _ _ _ _          |_  |         | |
.     | |     |  _ _| 12 _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |      _|   |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|7   _ _| |     | |     | |
.   | |     | |    4    |_                 _|  _ _|     | |     | |
.   | |     |_|_ _        |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| | 15 _|    _ _|     | |
.   |_|_ _ _        |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
.  8      | |_ _      |                       |      _|   |  _ _ _|
.         |_    |     |_ _ _ _ _ _            |  _ _|28  _| |
.           |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |      _|  _|
.          8  |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|
.                 |                               |  _ _|  31
.                 |_ _ _ _ _ _ _ _                | |
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.
.
[For two other drawings of the spiral see the links. - _N. J. A. Sloane_, Nov 16 2020]
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:
---------------------------------------------------------------
Triangle         Diagram of the symmetry of sigma (n = 1..24)
---------------------------------------------------------------
.              _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
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;           |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|* * * * * * *
10, 10;       |_ _ _ _ _ _ _ _ _ _| | |       |* * * * * * * *
42;           |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|* * * * * * * *
11, 5, 5, 11; |_ _ _ _ _ _ _ _ _ _ _| | |* * * * * * * * * * *
18, 18;       |_ _ _ _ _ _ _ _ _ _ _ _| |* * * * * * * * * * *
12, 12;       |_ _ _ _ _ _ _ _ _ _ _ _| |* * * * * * * * * * *
60;           |_ _ _ _ _ _ _ _ _ _ _ _ _|* * * * * * * * * * *
...
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).
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].
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.
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.
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.
Note that the number of *'s in the diagram is 24^2 - A024916(24) = 576 - 491 = A004125(24) = 85.
From _Omar E. Pol_, Nov 22 2020: (Start)
Also consider the infinite double-staircases diagram defined in A335616 (see the theorem).
For n = 15 the diagram with first 15 levels looks like this:
.
Level                         "Double-staircases" diagram
.                                          _
1                                        _|1|_
2                                      _|1 _ 1|_
3                                    _|1  |1|  1|_
4                                  _|1   _| |_   1|_
5                                _|1    |1 _ 1|    1|_
6                              _|1     _| |1| |_     1|_
7                            _|1      |1  | |  1|      1|_
8                          _|1       _|  _| |_  |_       1|_
9                        _|1        |1  |1 _ 1|  1|        1|_
10                     _|1         _|   | |1| |   |_         1|_
11                   _|1          |1   _| | | |_   1|          1|_
12                 _|1           _|   |1  | |  1|   |_           1|_
13               _|1            |1    |  _| |_  |    1|            1|_
14             _|1             _|    _| |1 _ 1| |_    |_             1|_
15            |1              |1    |1  | |1| |  1|    1|              1|
.
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:
.
Level                             "Ziggurat" diagram
.                                          _
6                                         |1|
7                            _            | |            _
8                          _|1|          _| |_          |1|_
9                        _|1  |         |1   1|         |  1|_
10                     _|1    |         |     |         |    1|_
11                   _|1      |        _|     |_        |      1|_
12                 _|1        |       |1       1|       |        1|_
13               _|1          |       |         |       |          1|_
14             _|1            |      _|    _    |_      |            1|_
15            |1              |     |1    |1|    1|     |              1|
.
The 15th row
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]
The 15th row
of triangle:  [              8,            8,            8              ]
The 15th row
of A296508:   [              8,      7,    1,    0,      8              ]
The 15th row
of A280851    [              8,      7,    1,            8              ]
.
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.
For the definition of subparts see A239387 and also A296508, A280851. (End)

Robert Price, Table of n, a(n) for n = 1..15542 (rows n = 1..5000, flattened)
Hartmut F. W. Hoft, Sample visual documentation for Mathematica code
Michel Marcus, A colored version of the symmetric representation of sigma(n), multipage, n = 1..85
Omar E. Pol, An infinite stepped pyramid (A237593, A237270, A262626)
Omar E. Pol, Perspective view of the stepped pyramid (16 levels)
Omar E. Pol, Perspective view of the stepped pyramid into four quadrants (11 levels). This is formed by combing four copies of the pyramid back-to-back (cf. A244050).
N. J. A. Sloane, Another drawing of the spiral
N. J. A. Sloane, Spiral showing only the outer boundary
Index entries for sequences related to sigma(n)

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, A384149.

T[n_,k_] := Ceiling[(n + 1)/k - (k + 1)/2] (* from A235791 *)
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];
rd = Transpose[{r, d}]; For[k = 1, k <= 2c, k++, p = Join[p, Map[Last[p] + rd[[k, 2]] * # &, Range[rd[[k, 1]]]]]]; p]
segments[n_] := SplitBy[Map[Min, Drop[Drop[path[n], 1], -1] - path[n - 1]], # == 0 &]
a237270[n_] := Select[Map[Apply[Plus, #] &, segments[n]], # != 0 &]
Flatten[Map[a237270, Range[40]]] (* data *)
(* Hartmut F. W. Hoft, Jun 23 2014 *)

T(n, k) = (A384149(n, k) + A384149(n, m+1-k))/2, where m = A237271(n) is the row length. (conjectured) - Peter Munn, Jun 01 2025

Drawing of the spiral extended by Omar E. Pol, Nov 22 2020

A239660 Triangle read by rows in which row n lists two copies of the n-th row of triangle A237593.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4, 4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4, 5, 2, 1, 1, 2, 5, 5, 2, 1, 1, 2, 5, 5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 5, 6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6, 6, 3, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1, 3, 6

Offset: 1

Omar E. Pol, Mar 24 2014

For the construction of this sequence also we can start from A235791.
This sequence can be interpreted as an infinite Dyck path: UDUDUUDD...
Also we use this sequence for the construction of a spiral in which the arms in the quadrants give the symmetric representation of sigma, see example.
We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016
The spiral has the property that the sum of the parts in the quadrants 1 and 3, divided by the sum of the parts in the quadrants 2 and 4, converges to 3/5. - Omar E. Pol, Jun 10 2019

			Triangle begins (first 15.5 rows):
1, 1, 1, 1;
2, 2, 2, 2;
2, 1, 1, 2, 2, 1, 1, 2;
3, 1, 1, 3, 3, 1, 1, 3;
3, 2, 2, 3, 3, 2, 2, 3;
4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4;
4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4;
5, 2, 1, 1, 2, 5, 5, 2, 1, 1, 2, 5;
5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 5;
6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6;
6, 3, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1, 3, 6;
7, 2, 2, 1, 1, 2, 2, 7, 7, 2, 2, 1, 1, 2, 2, 7;
7, 3, 2, 1, 1, 2, 3, 7, 7, 3, 2, 1, 1, 2, 3, 7;
8, 3, 1, 2, 2, 1, 3, 8, 8, 3, 1, 2, 2, 1, 3, 8;
8, 3, 2, 1, 1, 1, 1, 2, 3, 8, 8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
9, 3, 2, 1, 1, 1, 1, 2, 3, 9, ...
.
Illustration of initial terms as an infinite Dyck path (row n = 1..4):
.
.                            /\/\    /\/\
.       /\  /\  /\/\  /\/\  /    \  /    \
.  /\/\/  \/  \/    \/    \/      \/      \
.
.
Illustration of initial terms for the construction of a spiral related to sigma:
.
.  row 1     row 2          row 3           row 4
.                                          _ _ _
.                                               |_
.             _ _                                 |
.   _ _      |                                    |
.  |   |     |                                    |
.            |         |           |              |
.            |_ _      |_         _|              |
.                        |_ _ _ _|               _|
.                                          _ _ _|
.
.[1,1,1,1] [2,2,2,2] [2,1,1,2,2,1,1,2] [3,1,1,3,3,1,1,3]
.
The first 2*A003056(n) terms of the n-th row are represented in the A010883(n-1) quadrant and the last 2*A003056(n) terms of the n-th row are represented in the A010883(n) quadrant.
.
Illustration of the spiral constructed with the first 15.5 rows of triangle:
.
.               12 _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.                _| |                           |
.               |_ _|9 _ _ _ _ _ _              |_ _
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_
.      _ _ _| |      _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |      _ _| |   12 _ _ _ _          |_  |         | |
.     | |     |  _ _|    _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |      _|   |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|    _ _| |     | |     | |
.   | |     | |    4    |_               7 _|  _ _|     | |     | |
.   | |     |_|_ _        |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| |    _|    _ _|     | |
.   |_|_ _ _        |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
.  8      | |_ _      |                     15|      _|   |  _ _ _|
.         |_    |     |_ _ _ _ _ _            |  _ _|    _| |
.        8  |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |      _|  _|
.             |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|
.                 |                             28|  _ _|
.                 |_ _ _ _ _ _ _ _                | |
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.                                                    31
.
The diagram contains A237590(16) = 27 parts.
The total area (also the total number of cells) in the n-th arm of the spiral is equal to sigma(n) = A000203(n), considering every quadrant and the axes x and y. (checked by hand up to row n = 128). The parts of the spiral are in A237270: 1, 3, 2, 2, 7...
Diagram extended by _Omar E. Pol_, Aug 23 2018

Robert Price, Table of n, a(n) for n = 1..30016 (rows n = 1..412, flattened)

Row n has length 4*A003056(n).
The sum of row n is equal to 4*n = A008586(n).
Row n is a palindromic composition of 4*n = A008586(n).
Both column 1 and right border are A008619, n >= 1.
The connection between A196020 and A237270 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> this sequence --> A237270.
Cf. A000203, A000217, A003056, A008619, A010883, A112610, A193553, A237048, A237271, A237590, A239052, A239053, A239663, A239665, A239931, A239932, A239933, A239934, A240020, A240062, A244050, A245092, A262626, A296508, A299778.

A239665 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma of the smallest number whose symmetric representation of sigma has n parts.

Original entry on oeis.org

1, 2, 2, 5, 3, 5, 11, 5, 5, 11, 32, 12, 16, 12, 32, 74, 26, 14, 14, 26, 74, 179, 61, 29, 38, 29, 61, 179, 452, 152, 68, 32, 32, 68, 152, 452, 1250, 418, 182, 152, 100, 152, 182, 418, 1250, 3035, 1013, 437, 342, 85, 85, 342, 437, 1013, 3035, 6958, 1394, 638, 314, 154, 236, 154, 314, 638, 1394, 6958

Offset: 1

Omar E. Pol, Mar 23 2014

Row n is also row A239663(n) of A237270.

			----------------------------------------------------------------------
n    A239663(n)  Triangle begins:                        A266094(n)
----------------------------------------------------------------------
1        1       [1]                                         1
2        3       [2, 2]                                      4
3        9       [5, 3, 5]                                  13
4       21       [11, 5, 5, 11]                             32
5       63       [32, 12, 16, 12, 32]                      104
6      147       [74, 26, 14, 14, 26, 74]                  228
7      357       [179, 61, 29, 38, 29, 61, 179]            576
8      903       [452, 152, 68, 32, 32, 68, 152, 452]     1408
...
Illustration of initial terms:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.     _ _ 2          | |
.    |_ _|_ 2        | |
.     _ 1| |         | |
.    |_| |_|         |_|
.
For n = 2 we have that A239663(2) = 3 is the smallest number whose symmetric representation of sigma has 2 parts. Row 3 of A237593 is [2, 1, 1, 2] and row 2 of A237593 is [2, 2] therefore between both Dyck paths in the first quadrant there are two regions (or parts) of sizes [2, 2], so row 2 is [2, 2].
For n = 3 we have that A239663(3) = 9 is the smallest number whose symmetric representation of sigma has 3 parts. 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 Dyck paths in the first quadrant there are three regions (or parts) of sizes [5, 3, 5], so row 3 is [5, 3, 5].

Cf. A000203, A005279, A196020, A236104, A237270, A237271, A235791, A237591, A237593, A239660, A239663, A239931-A239934, A240020, A240062, A244050, A245092, A262626, A266094.

a(16)-a(28) from Michel Marcus and Omar E. Pol, Mar 28 2014
a(29)-a(36) from Michel Marcus, Mar 28 2014
a(37)-a(45) from Michel Marcus, Mar 29 2014
a(46)-a(66) from Michel Marcus, Apr 02 2014

A252922 a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3), with a(1)=0, a(2)=1, a(3)=4.

Original entry on oeis.org

0, 1, 4, 8, 14, 17, 25, 26, 35, 36, 46, 43, 58, 54, 66, 62, 79, 73, 88, 77, 101, 94, 110, 92, 120, 115, 133, 113, 138, 126, 158, 134, 167, 143, 165, 150, 193, 177, 189, 154, 206, 188, 228, 182, 224, 206, 234, 198, 244, 229, 274, 222, 263, 224, 272, 246, 312, 272, 290, 230, 318, 290, 326, 262, 327, 315, 355, 296

Offset: 1

Omar E. Pol, Dec 24 2014

This is also a rectangular array read by rows, with four columns, in which T(j,k) is the number of cells (also the area) of the j-th gap between the arms in the k-th quadrant of the spiral of the symmetric representation of sigma described in A239660, with j >= 1 and 1 <= k <= 4 and starting with T(1,1) = 0, see example.

We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

			a(5) = sigma(4) + sigma(3) + sigma(2) = 7 + 4 + 3 = 14. On the other hand a(5) = A024916(4) - A024916(1) = 15 - 1 = 14.
...
Also, if written as a rectangular array T(j,k) with four columns the sequence begins:
    0,   1,   4,   8;
   14,  17,  25,  26;
   35,  36,  46,  43;
   58,  54,  66,  62;
   79,  73,  88,  77;
  101,  94, 110,  92;
  120, 115, 133, 113;
  138, 126, 158, 134;
  167, 143, 165, 150;
  193, 177, 189, 154;
  206, 188, 228, 182;
  224, 206, 234, 198;
  244, 229, 274, 222;
  263, 224, 272, 246;
  312, 272, 290, 230;
  318, 290, 326, 262;
  ...
In this case T(2,1) = a(5) = 14.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A010883, A024916, A092403, A112610, A193553, A196020, A236104, A237270, A237271, A237593, A239052, A239053, A239931-A239934, A239660, A240020, A244050, A245092, A262626.

L:= [0,0,0,seq(numtheory:-sigma(n), n=1..100)]:
L[1..101]+L[2..102]+L[3..103]; # Robert Israel, Dec 07 2016

a252922[n_] := Block[{f}, f[1] = 0; f[2] = 1; f[3] = 4;
  f[x_] := DivisorSigma[1, x - 1] + DivisorSigma[1, x - 2] +
DivisorSigma[1, x - 3]; Table[f[i], {i, n}]]; a252922[68] (* Michael De Vlieger, Dec 27 2014 *)

v=concat([0,1,4],vector(100,n,sigma(n)+sigma(n+1)+sigma(n+2))) \\ Derek Orr, Dec 30 2014

a(1) = 0, a(2) = sigma(1) = 1, a(3) = sigma(2) + sigma(1) = 4; for n >= 4, a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3).

a(n) = A024916(n-1) - A024916(n-4) for n >= 5.

cp's OEIS Frontend

A237270 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(n).

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A239660 Triangle read by rows in which row n lists two copies of the n-th row of triangle A237593.

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A239665 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma of the smallest number whose symmetric representation of sigma has n parts.

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A252922 a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3), with a(1)=0, a(2)=1, a(3)=4.

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