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1, 1, 2, 2, 3, 3, 2, 1, 4, 5, 3, 1, 5, 7, 2, 1, 4, 2, 3, 1, 6, 11, 3, 1, 5, 2, 4, 1, 7, 15, 2, 1, 4, 2, 3, 1, 6, 4, 5, 1, 4, 1, 8, 22, 3, 1, 5, 2, 4, 1, 7, 4, 3, 1, 6, 2, 5, 1, 9, 30, 2, 1, 4, 2, 3, 1, 6, 4, 5, 1, 4, 1, 8, 7, 4, 1, 7, 2, 6, 1, 5, 1, 10, 42
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OFFSET
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1,3
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COMMENTS
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Number of toothpicks added at n-th stage to the toothpick structure (related to integer partitions) of A225600.
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LINKS
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FORMULA
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EXAMPLE
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Written as an irregular triangle in which row n has length 2*A187219(n) we can see that the right border gives A000041 and the previous term of the last term in row n is n.
1,1;
2,2;
3,3;
2,1,4,5;
3,1,5,7;
2,1,4,2,3,1,6,11;
3,1,5,2,4,1,7,15;
2,1,4,2,3,1,6,4,5,1,4,1,8,22;
3,1,5,2,4,1,7,4,3,1,6,2,5,1,9,30;
2,1,4,2,3,1,6,4,5,1,4,1,8,7,4,1,7,2,6,1,5,1,10,42;
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Illustration of the first seven rows of triangle as a minimalist diagram of regions of the set of partitions of 7:
. _ _ _ _ _ _ _
. 15 _ _ _ _ |
. _ _ _ _|_ |
. _ _ _ | |
. _ _ _|_ _|_ |
. 11 _ _ _ | |
. _ _ _|_ | |
. _ _ | | |
. _ _|_ _|_ | |
. 7 _ _ _ | | |
. _ _ _|_ | | |
. 5 _ _ | | | |
. _ _|_ | | | |
. 3 _ _ | | | | |
. 2 _ | | | | | |
. 1 | | | | | | |
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. 1 2 3 4 5 6 7
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Also using the elements of this diagram we can draw a Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See below:
.
7..................................
. /\
5.................... / \ /\
. /\ / \ /\ /
3.......... / \ / \ / \/
2..... /\ / \ /\/ \ /
1.. /\ / \ /\/ \ / \ /\/
0 /\/ \/ \/ \/ \/
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CROSSREFS
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Cf. A000041, A006128, A135010, A138137, A141285, A179862, A186114, A186412, A187219, A194446, A206437, A211978, A220517, A225600, A225610.
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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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