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A332266
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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.
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4
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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
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OFFSET
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1,2
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COMMENTS
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This triangle can be interpreted as a table of partitions into consecutive parts that differ by 2 (see the Example section).
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LINKS
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EXAMPLE
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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.
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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.
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Illustration of initial terms arranged into a triangular structure:
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. _|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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CROSSREFS
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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).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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