%I #33 Dec 21 2020 07:29:40
%S 1,2,3,4,1,5,3,6,2,7,4,8,3,9,5,1,10,4,3,11,6,5,12,5,2,13,7,4,14,6,6,
%T 15,8,3,16,7,5,1,17,9,7,3,18,8,4,5,19,10,6,7,20,9,8,2,21,11,5,4,22,10,
%U 7,6,23,12,9,8,24,11,6,3,25,13,8,5,1,26,12,10,7,3,27,14,7,9,5
%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 2, where the m-th block starts with m, m >= 1, and the first element of column k is in row k^2.
%C This triangle can be interpreted as a table of partitions into consecutive parts that differ by 2 (see the Example section).
%e Triangle begins:
%e 1;
%e 2;
%e 3;
%e 4, 1;
%e 5, 3;
%e 6, 2;
%e 7, 4;
%e 8, 3;
%e 9, 5, 1;
%e 10, 4, 3;
%e 11, 6, 5;
%e 12, 5, 2;
%e 13, 7, 4;
%e 14, 6, 6;
%e 15, 8, 3;
%e 16, 7, 5, 1;
%e 17, 9, 7, 3;
%e 18, 8, 4, 5;
%e 19, 10, 6, 7;
%e 20, 9, 8, 2;
%e 21, 11, 5, 4;
%e 22, 10, 7, 6;
%e 23, 12, 9, 8;
%e 24, 11, 6, 3;
%e 25, 13, 8, 5, 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 2:
%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],[1];| 4, 1;| 4, 1; | 4, 1;|
%e 5 | | | | 5, [3];| [5], 3;| 5, 3; | 5, 3;|
%e 6 | | | | | | [6],[2];| 6, 2;|
%e 7 | | | | | | 7, [4];| [7], 4;|
%e . ---------------------------------------------------------
%e Figure F: for n = 6 the partitions of 6 into consecutive parts that differ by 2 (but with the parts in increasing order) are [6] and [2, 4]. 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 6.
%e .
%e Figures H..L show the location (in the columns of the table) of the partitions of 8..12 (respectively) into consecutive parts that differ by 2:
%e . -----------------------------------------------------------------------
%e Fig: H I J K L
%e . -----------------------------------------------------------------------
%e . n: 8 9 10 11 12
%e Row -----------------------------------------------------------------------
%e 1 | 1; | 1; | 1; | 1; | 1; |
%e 1 | 2; | 2; | 2; | 2; | 2; |
%e 3 | 3; | 3; | 3; | 3; | 3; |
%e 4 | 4, 1; | 4, 1; | 4, 1; | 4, 1; | 4, 1; |
%e 5 | 5, 3; | 5, 3; | 5, 3; | 5, 3; | 5, 3; |
%e 6 | 6, 2; | 6, 2; | 6, 2; | 6, 2; | 6, 2; |
%e 7 | 7, 4; | 7, 4; | 7, 4; | 7, 4; | 7, 4; |
%e 8 | [8],[3]; | 8, 3; | 8, 3; | 8, 3; | 8, 3; |
%e 9 | 9, [5], 1;| [9], 5, [1];| 9, 5, 1;| 9, 5, 1;| 9, 5, 1; |
%e 10 | | 10, 4, [3];| [10],[4], 3;| 10, 4, 3;| 10, 4; 3; |
%e 11 | | 11, 6, [5];| 11, [6], 5;| [11], 6, 5,| 11, 6; 5; |
%e 12 | | | | | [12],[5],[2];|
%e 13 | | | | | 13, [7],[4];|
%e 14 | | | | | 14, 6, [6];|
%e . -----------------------------------------------------------------------
%e Figure I: for n = 9 the partitions of 9 into consecutive parts that differ by 2(but with the parts in increasing order) are [9] and [1, 3, 5]. These partitions have one part and three parts respectively. On the other hand, we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 9.
%e .
%e Illustration of initial terms arranged into a triangular structure:
%e . _
%e . _|1|
%e . _|2 |
%e . _|3 _|
%e . _|4 |1|
%e . _|5 _|3|
%e . _|6 |2 |
%e . _|7 _|4 |
%e . _|8 |3 _|
%e . _|9 _|5 |1|
%e . _|10 |4 |3|
%e . _|11 _|6 _|5|
%e . _|12 |5 |2 |
%e . _|13 _|7 |4 |
%e . _|14 |6 _|6 |
%e . _|15 _|8 |3 _|
%e . _|16 |7 |5 |1|
%e . _|17 _|9 _|7 |3|
%e . _|18 |8 |4 |5|
%e . _|19 _|10 |6 _|7|
%e . _|20 |9 _|8 |2 |
%e . _|21 _|11 |5 |4 |
%e . _|22 |10 |7 |6 |
%e . _|23 _|12 _|9 _|8 |
%e . _|24 |11 |6 |3 _|
%e . |25 |13 |8 |5 |1|
%e ...
%e The number of horizontal line segments in the n-th row of the diagram equals A038548(n), the number of partitions of n into consecutive parts that differ by 2.
%Y Tables of the same family where the consecutive parts differ by d are A010766 (d=0), A286001 (d=1), this sequence (d=2), A334945 (d=3), A334618(d=4).
%Y Cf. A038548, A060872, A066839, A303300, A330466.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Feb 08 2020