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A334618 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists successive blocks of k consecutive integers that differ by 4, where the m-th block starts with m, m >= 1, and the first element of column k is in the row that is the k-th hexagonal number (A000384).

%I A334618 #54 Dec 21 2020 07:29:48
%S A334618 1,2,3,4,5,6,1,7,5,8,2,9,6,10,3,11,7,12,4,13,8,14,5,15,9,1,16,6,5,17,
%T A334618 10,9,18,7,2,19,11,6,20,8,10,21,12,3,22,9,7,23,13,11,24,10,4,25,14,8,
%U A334618 26,11,12,27,15,5,28,12,9,1,29,16,13,5,30,13,6,9,31,17,10,13,32,14,14,2
%N A334618 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists successive blocks of k consecutive integers that differ by 4, where the m-th block starts with m, m >= 1, and the first element of column k is in the row that is the k-th hexagonal number (A000384).
%C A334618 This triangle can be interpreted as a table of partitions into consecutive parts that differ by 4 (see the Example section).
%C A334618 Also, every triangle of this family has the property that starting from row n the sum of k positive and consecutive terms in the column k is equal to n.
%e A334618 Triangle begins (rows 1..28):
%e A334618    1;
%e A334618    2;
%e A334618    3;
%e A334618    4;
%e A334618    5;
%e A334618    6,  1;
%e A334618    7,  5;
%e A334618    8,  2;
%e A334618    9,  6;
%e A334618   10,  3;
%e A334618   11,  7;
%e A334618   12,  4;
%e A334618   13,  8;
%e A334618   14,  5;
%e A334618   15,  9,  1;
%e A334618   16,  6,  5;
%e A334618   17, 10,  9;
%e A334618   18,  7,  2;
%e A334618   19, 11,  6;
%e A334618   20,  8, 10;
%e A334618   21, 12,  3;
%e A334618   22,  9,  7;
%e A334618   23, 13, 11;
%e A334618   24, 10,  4;
%e A334618   25, 14,  8;
%e A334618   26, 11, 12;
%e A334618   27, 15,  5;
%e A334618   28, 12,  9,  1;
%e A334618 ...
%e A334618 Figures A..H show the location (in the columns of the table) of the partitions of n = 1..8 (respectively) into consecutive parts that differ by 4:
%e A334618 .   -----------------------------------------------------------
%e A334618 Fig:   A     B     C     D     E       F        G        H
%e A334618 .   -----------------------------------------------------------
%e A334618 . n:   1     2     3     4     5       6        7        8
%e A334618 Row -----------------------------------------------------------
%e A334618 1   | [1];|  1; |  1; |  1; |  1; |  1;     |  1;   |  1;     |
%e A334618 2   |     | [2];|  2; |  2; |  2; |  2;     |  2;   |  2;     |
%e A334618 3   |     |     | [3];|  3; |  3; |  3;     |  3;   |  3;     |
%e A334618 4   |     |     |     | [4];|  4; |  4;     |  4;   |  4;     |
%e A334618 5   |     |     |     |     | [5];|  5;     |  5;   |  5;     |
%e A334618 6   |     |     |     |     |     | [6],[1];|  6, 1;|  6,  1; |
%e A334618 7   |     |     |     |     |     |     [5];| [7],5;|  7,  5; |
%e A334618 8   |     |     |     |     |     |         |       | [8],[2];|
%e A334618 9   |     |     |     |     |     |         |       |  9, [6];|
%e A334618 .   -----------------------------------------------------------
%e A334618 Figure H: for n = 8 the partitions of 8 into consecutive parts that differ by 4 (but with the parts in increasing order) are [8] and [2, 6]. These partitions have one part and two parts respectively. On the other hand  we can find the mentioned partitions in the columns 1 and 2 of this table, starting at the row 8.
%e A334618 .
%e A334618 Illustration of initial terms arranged into a triangular structure:
%e A334618 .                                                           _
%e A334618 .                                                         _|1|
%e A334618 .                                                       _|2  |
%e A334618 .                                                     _|3    |
%e A334618 .                                                   _|4      |
%e A334618 .                                                 _|5       _|
%e A334618 .                                               _|6        |1|
%e A334618 .                                             _|7         _|5|
%e A334618 .                                           _|8          |2  |
%e A334618 .                                         _|9           _|6  |
%e A334618 .                                       _|10           |3    |
%e A334618 .                                     _|11            _|7    |
%e A334618 .                                   _|12             |4      |
%e A334618 .                                 _|13              _|8      |
%e A334618 .                               _|14               |5       _|
%e A334618 .                             _|15                _|9      |1|
%e A334618 .                           _|16                 |6        |5|
%e A334618 .                         _|17                  _|10      _|9|
%e A334618 .                       _|18                   |7        |2  |
%e A334618 .                     _|19                    _|11       |6  |
%e A334618 .                   _|20                     |8         _|10 |
%e A334618 .                 _|21                      _|12       |3    |
%e A334618 .               _|22                       |9          |7    |
%e A334618 .              |23                         |13         |11   |
%e A334618 ...
%e A334618 The number of horizontal line segments in the n-th row of the diagram equals A334461(n), the number of partitions of n into consecutive parts that differ by 4.
%Y A334618 Tables of the same family where the consecutive parts differ by d are A010766 (d=0), A286001 (d=1), A332266 (d=2), A334945 (d=3), this sequence (d=4).
%Y A334618 Cf. A000384, A327262, A334460, A334461, A334462, A334464.
%K A334618 nonn,tabf
%O A334618 1,2
%A A334618 _Omar E. Pol_, Dec 18 2020