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A334945
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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 3, 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 pentagonal number (A000326).
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4
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1, 2, 3, 4, 5, 1, 6, 4, 7, 2, 8, 5, 9, 3, 10, 6, 11, 4, 12, 7, 1, 13, 5, 4, 14, 8, 7, 15, 6, 2, 16, 9, 5, 17, 7, 8, 18, 10, 3, 19, 8, 6, 20, 11, 9, 21, 9, 4, 22, 12, 7, 1, 23, 10, 10, 4, 24, 13, 5, 7, 25, 11, 8, 10, 26, 14, 11, 2, 27, 12, 6, 5, 28, 15, 9, 8, 29, 13, 12, 11, 30, 16, 7, 3
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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 3 (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. - Omar E. Pol, Dec 18 2020
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LINKS
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EXAMPLE
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Triangle begins:
1;
2;
3;
4;
5, 1;
6, 4;
7, 2;
8, 5;
9, 3;
10, 6;
11, 4;
12, 7, 1;
13, 5, 4;
14, 8, 7;
15, 6, 2;
16, 9, 5;
17, 7, 8;
18, 10, 3;
19, 8, 6;
20, 11, 9;
21, 9, 4;
22, 12, 7, 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 3:
. -----------------------------------------------------
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];| 4; | 4; | 4; |
5 | | | | | [5],[1];| 5, 1;| 5, 1; |
6 | | | | | 6, [4];| [6],4;| 6, 4; |
7 | | | | | | | [7],[2];|
8 | | | | | | | 8, [5];|
. -----------------------------------------------------
Figure G: for n = 7 the partitions of 7 into consecutive parts that differ by 3 (but with the parts in increasing order) are [7] and [2, 5]. 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 7.
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Illustration of initial terms arranged into a triangular structure:
. _
. _|1|
. _|2 |
. _|3 |
. _|4 _|
. _|5 |1|
. _|6 _|4|
. _|7 |2 |
. _|8 _|5 |
. _|9 |3 |
. _|10 _|6 |
. _|11 |4 _|
. _|12 _|7 |1|
. _|13 |5 |4|
. _|14 _|8 _|7|
. _|15 |6 |2 |
. _|16 _|9 |5 |
. _|17 |7 _|8 |
. _|18 _|10 |3 |
. _|19 |8 |6 |
. _|20 _|11 _|9 |
. _|21 |9 |4 _|
. |22 |12 |7 |1|
...
The number of horizontal line segments in the n-th row of the diagram equals A117277(n), the number of partitions of n into consecutive parts that differ by 3.
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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), this sequence (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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