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A334618
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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 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).
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4
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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, 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, 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
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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 4 (see the Example section).
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.
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LINKS
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EXAMPLE
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Triangle begins (rows 1..28):
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, 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;
26, 11, 12;
27, 15, 5;
28, 12, 9, 1;
...
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:
. -----------------------------------------------------------
Fig: A B C D E F G H
. -----------------------------------------------------------
. n: 1 2 3 4 5 6 7 8
Row -----------------------------------------------------------
1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; | 1; |
2 | | [2];| 2; | 2; | 2; | 2; | 2; | 2; |
3 | | | [3];| 3; | 3; | 3; | 3; | 3; |
4 | | | | [4];| 4; | 4; | 4; | 4; |
5 | | | | | [5];| 5; | 5; | 5; |
6 | | | | | | [6],[1];| 6, 1;| 6, 1; |
7 | | | | | | [5];| [7],5;| 7, 5; |
8 | | | | | | | | [8],[2];|
9 | | | | | | | | 9, [6];|
. -----------------------------------------------------------
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.
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Illustration of initial terms arranged into a triangular structure:
. _
. _|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 _|10 _|9|
. _|18 |7 |2 |
. _|19 _|11 |6 |
. _|20 |8 _|10 |
. _|21 _|12 |3 |
. _|22 |9 |7 |
. |23 |13 |11 |
...
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.
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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), A332266 (d=2), A334945 (d=3), this sequence (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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