%I #54 Dec 21 2020 07:29:48
%S 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 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 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 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 This triangle can be interpreted as a table of partitions into consecutive parts that differ by 4 (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.
%e Triangle begins (rows 1..28):
%e 1;
%e 2;
%e 3;
%e 4;
%e 5;
%e 6, 1;
%e 7, 5;
%e 8, 2;
%e 9, 6;
%e 10, 3;
%e 11, 7;
%e 12, 4;
%e 13, 8;
%e 14, 5;
%e 15, 9, 1;
%e 16, 6, 5;
%e 17, 10, 9;
%e 18, 7, 2;
%e 19, 11, 6;
%e 20, 8, 10;
%e 21, 12, 3;
%e 22, 9, 7;
%e 23, 13, 11;
%e 24, 10, 4;
%e 25, 14, 8;
%e 26, 11, 12;
%e 27, 15, 5;
%e 28, 12, 9, 1;
%e ...
%e 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 . -----------------------------------------------------------
%e Fig: A B C D E F G H
%e . -----------------------------------------------------------
%e . n: 1 2 3 4 5 6 7 8
%e Row -----------------------------------------------------------
%e 1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; | 1; |
%e 2 | | [2];| 2; | 2; | 2; | 2; | 2; | 2; |
%e 3 | | | [3];| 3; | 3; | 3; | 3; | 3; |
%e 4 | | | | [4];| 4; | 4; | 4; | 4; |
%e 5 | | | | | [5];| 5; | 5; | 5; |
%e 6 | | | | | | [6],[1];| 6, 1;| 6, 1; |
%e 7 | | | | | | [5];| [7],5;| 7, 5; |
%e 8 | | | | | | | | [8],[2];|
%e 9 | | | | | | | | 9, [6];|
%e . -----------------------------------------------------------
%e 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 .
%e Illustration of initial terms arranged into a triangular structure:
%e . _
%e . _|1|
%e . _|2 |
%e . _|3 |
%e . _|4 |
%e . _|5 _|
%e . _|6 |1|
%e . _|7 _|5|
%e . _|8 |2 |
%e . _|9 _|6 |
%e . _|10 |3 |
%e . _|11 _|7 |
%e . _|12 |4 |
%e . _|13 _|8 |
%e . _|14 |5 _|
%e . _|15 _|9 |1|
%e . _|16 |6 |5|
%e . _|17 _|10 _|9|
%e . _|18 |7 |2 |
%e . _|19 _|11 |6 |
%e . _|20 |8 _|10 |
%e . _|21 _|12 |3 |
%e . _|22 |9 |7 |
%e . |23 |13 |11 |
%e ...
%e 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 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 Cf. A000384, A327262, A334460, A334461, A334462, A334464.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Dec 18 2020