%I #26 Dec 19 2020 07:56:27
%S 1,2,3,4,5,1,6,4,7,2,8,5,9,3,10,6,11,4,12,7,1,13,5,4,14,8,7,15,6,2,16,
%T 9,5,17,7,8,18,10,3,19,8,6,20,11,9,21,9,4,22,12,7,1,23,10,10,4,24,13,
%U 5,7,25,11,8,10,26,14,11,2,27,12,6,5,28,15,9,8,29,13,12,11,30,16,7,3
%N 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 3, 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 pentagonal number (A000326).
%C This triangle can be interpreted as a table of partitions into consecutive parts that differ by 3 (see the Example section).
%C 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. - _Omar E. Pol_, Dec 18 2020
%e Triangle begins:
%e 1;
%e 2;
%e 3;
%e 4;
%e 5, 1;
%e 6, 4;
%e 7, 2;
%e 8, 5;
%e 9, 3;
%e 10, 6;
%e 11, 4;
%e 12, 7, 1;
%e 13, 5, 4;
%e 14, 8, 7;
%e 15, 6, 2;
%e 16, 9, 5;
%e 17, 7, 8;
%e 18, 10, 3;
%e 19, 8, 6;
%e 20, 11, 9;
%e 21, 9, 4;
%e 22, 12, 7, 1;
%e ...
%e Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts that differ by 3:
%e . -----------------------------------------------------
%e Fig: A B C D E F G
%e . -----------------------------------------------------
%e . n: 1 2 3 4 5 6 7
%e Row -----------------------------------------------------
%e 1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; |
%e 2 | | [2];| 2; | 2; | 2; | 2; | 2; |
%e 3 | | | [3];| 3; | 3; | 3; | 3; |
%e 4 | | | | [4];| 4; | 4; | 4; |
%e 5 | | | | | [5],[1];| 5, 1;| 5, 1; |
%e 6 | | | | | 6, [4];| [6],4;| 6, 4; |
%e 7 | | | | | | | [7],[2];|
%e 8 | | | | | | | 8, [5];|
%e . -----------------------------------------------------
%e Figure G: for n = 7 the partitions of 7 into consecutive parts that differ by 3 (but with the parts in increasing order) are [7] and [2, 5]. 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 7.
%e .
%e Illustration of initial terms arranged into a triangular structure:
%e . _
%e . _|1|
%e . _|2 |
%e . _|3 |
%e . _|4 _|
%e . _|5 |1|
%e . _|6 _|4|
%e . _|7 |2 |
%e . _|8 _|5 |
%e . _|9 |3 |
%e . _|10 _|6 |
%e . _|11 |4 _|
%e . _|12 _|7 |1|
%e . _|13 |5 |4|
%e . _|14 _|8 _|7|
%e . _|15 |6 |2 |
%e . _|16 _|9 |5 |
%e . _|17 |7 _|8 |
%e . _|18 _|10 |3 |
%e . _|19 |8 |6 |
%e . _|20 _|11 _|9 |
%e . _|21 |9 |4 _|
%e . |22 |12 |7 |1|
%e ...
%e The number of horizontal line segments in the n-th row of the diagram equals A117277(n), the number of partitions of n into consecutive parts that differ by 3.
%Y Tables of the same family where the consecutive parts differ by d are A010766 (d=0), A286001 (d=1), A332266 (d=2), this sequence (d=3), A334618(d=4).
%Y Cf. A000326, A117277, A330887, A330888, A330889, A334463.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, May 27 2020