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A347579
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Irregular triangle read by rows: T(n,k) = 1 iff there is no partition of n into exactly k consecutive parts, for n >= 1, 1 <= k <= A003056(n).
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2
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0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1
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OFFSET
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1
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COMMENTS
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins (rows 1..28):
0;
0;
0, 0;
0, 1;
0, 0;
0, 1, 0;
0, 0, 1;
0, 1, 1;
0, 0, 0;
0, 1, 1, 0;
0, 0, 1, 1;
0, 1, 0, 1;
0, 0, 1, 1;
0, 1, 1, 0;
0, 0, 0, 1, 0;
0, 1, 1, 1, 1;
0, 0, 1, 1, 1;
0, 1, 0, 0, 1;
0, 0, 1, 1, 1;
0, 1, 1, 1, 0;
0, 0, 0, 1, 1, 0;
0, 1, 1, 0, 1, 1;
0, 0, 1, 1, 1, 1;
0, 1, 0, 1, 1, 1;
0, 0, 1, 1, 0, 1;
0, 1, 1, 0, 1, 1;
0, 0, 0, 1, 1, 0;
0, 1, 1, 1, 1, 1, 0;
...
For n = 15 the partitions of 15 into consecutive parts are [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1]. We can see that a partition of 15 into four consecutive parts does not exist, hence there is a "nonexistence" of such partition, so T(15,4) = 1.
Illustration of initial terms:
Row _
1 _|0|
2 _|0 _|
3 _|0 |0|
4 _|0 _|1|
5 _|0 |0 _|
6 _|0 _|1|0|
7 _|0 |0 |1|
8 _|0 _|1 _|1|
9 _|0 |0 |0 _|
10 _|0 _|1 |1|0|
11 _|0 |0 _|1|1|
12 _|0 _|1 |0 |1|
13 _|0 |0 |1 _|1|
14 _|0 _|1 _|1|0 _|
15 _|0 |0 |0 |1|0|
16 _|0 _|1 |1 |1|1|
17 _|0 |0 _|1 _|1|1|
18 _|0 _|1 |0 |0 |1|
19 _|0 |0 |1 |1 _|1|
20 _|0 _|1 _|1 |1|0 _|
21 _|0 |0 |0 _|1|1|0|
22 _|0 _|1 |1 |0 |1|1|
23 _|0 |0 _|1 |1 |1|1|
24 _|0 _|1 |0 |1 _|1|1|
25 _|0 |0 |1 _|1|0 |1|
26 _|0 _|1 _|1 |0 |1 _|1|
27 _|0 |0 |0 |1 |1|0 _|
28 |0 |1 |1 |1 |1|1|0|
...
For the connection between the above diagram and the partitions of n into consecutive parts see the diagrams of A286000 and A286001.
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CROSSREFS
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The number of zeros in row n equals A001227(n).
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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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