A332266 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.
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, 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, 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
Offset: 1
Examples
Triangle begins: 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; 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, 7, 6; 23, 12, 9, 8; 24, 11, 6, 3; 25, 13, 8, 5, 1; ... 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: . --------------------------------------------------------- Fig: A B C D E F G . --------------------------------------------------------- . n: 1 2 3 4 5 6 7 Row --------------------------------------------------------- 1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; | 2 | | [2];| 2; | 2; | 2; | 2; | 2; | 3 | | | [3];| 3; | 3; | 3; | 3; | 4 | | | | [4],[1];| 4, 1;| 4, 1; | 4, 1;| 5 | | | | 5, [3];| [5], 3;| 5, 3; | 5, 3;| 6 | | | | | | [6],[2];| 6, 2;| 7 | | | | | | 7, [4];| [7], 4;| . --------------------------------------------------------- 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. . 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: . ----------------------------------------------------------------------- Fig: H I J K L . ----------------------------------------------------------------------- . n: 8 9 10 11 12 Row ----------------------------------------------------------------------- 1 | 1; | 1; | 1; | 1; | 1; | 1 | 2; | 2; | 2; | 2; | 2; | 3 | 3; | 3; | 3; | 3; | 3; | 4 | 4, 1; | 4, 1; | 4, 1; | 4, 1; | 4, 1; | 5 | 5, 3; | 5, 3; | 5, 3; | 5, 3; | 5, 3; | 6 | 6, 2; | 6, 2; | 6, 2; | 6, 2; | 6, 2; | 7 | 7, 4; | 7, 4; | 7, 4; | 7, 4; | 7, 4; | 8 | [8],[3]; | 8, 3; | 8, 3; | 8, 3; | 8, 3; | 9 | 9, [5], 1;| [9], 5, [1];| 9, 5, 1;| 9, 5, 1;| 9, 5, 1; | 10 | | 10, 4, [3];| [10],[4], 3;| 10, 4, 3;| 10, 4; 3; | 11 | | 11, 6, [5];| 11, [6], 5;| [11], 6, 5,| 11, 6; 5; | 12 | | | | | [12],[5],[2];| 13 | | | | | 13, [7],[4];| 14 | | | | | 14, 6, [6];| . ----------------------------------------------------------------------- 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. . Illustration of initial terms arranged into a triangular structure: . _ . _|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 | . _|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 |7 |6 | . _|23 _|12 _|9 _|8 | . _|24 |11 |6 |3 _| . |25 |13 |8 |5 |1| ... 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.
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