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A225600
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Toothpick sequence related to integer partitions (see Comments lines for definition).
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20
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0, 1, 2, 4, 6, 9, 12, 14, 15, 19, 24, 27, 28, 33, 40, 42, 43, 47, 49, 52, 53, 59, 70, 73, 74, 79, 81, 85, 86, 93, 108, 110, 111, 115, 117, 120, 121, 127, 131, 136, 137, 141, 142, 150, 172, 175, 176, 181, 183, 187, 188, 195, 199, 202, 203, 209, 211, 216, 217, 226, 256
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OFFSET
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0,3
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COMMENTS
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This infinite toothpick structure is a minimalist diagram of regions of the set of partitions of all positive integers. For the definition of "region" see A206437. The sequence shows the growth of the diagram as a cellular automaton in which the "input" is A141285 and the "output” is A194446.
To define the sequence we use the following rules:
We start in the first quadrant of the square grid with no toothpicks.
If n is odd we place A141285((n+1)/2) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the grid point (0, (n+1)/2).
If n is even we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. In this case the number of toothpicks added in vertical direction is equal to A194446(n/2).
The sequence gives the number of toothpicks after n stages. A220517 (the first differences) gives the number of toothpicks added at the n-th stage.
Also the toothpick structure (HV/HHVV/HHHVVV/HHV/HHHHVVVVV...) can be transformed in a Dyck path (UDUUDDUUUDDDUUDUUUUDDDDD...) 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, so the sequence can be represented by the vertices (or the number of steps from the origin) of the Dyck path. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See Example section. See also A211978, A220517, A225610.
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LINKS
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FORMULA
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EXAMPLE
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For n = 30 the structure has 108 toothpicks, so a(30) = 108.
. Diagram of regions
Partitions of 7 and partitions of 7
. _ _ _ _ _ _ _
7 15 _ _ _ _ |
4 + 3 _ _ _ _|_ |
5 + 2 _ _ _ | |
3 + 2 + 2 _ _ _|_ _|_ |
6 + 1 11 _ _ _ | |
3 + 3 + 1 _ _ _|_ | |
4 + 2 + 1 _ _ | | |
2 + 2 + 2 + 1 _ _|_ _|_ | |
5 + 1 + 1 7 _ _ _ | | |
3 + 2 + 1 + 1 _ _ _|_ | | |
4 + 1 + 1 + 1 5 _ _ | | | |
2 + 2 + 1 + 1 + 1 _ _|_ | | | |
3 + 1 + 1 + 1 + 1 3 _ _ | | | | |
2 + 1 + 1 + 1 + 1 + 1 2 _ | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1 1 | | | | | | |
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. 1 2 3 4 5 6 7
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Illustration of initial terms:
.
. _ _ _ _ _ _
. _ _ _ _ _ _ _ _ |
. _ _ _ _ | _ | _ | |
. | | | | | | | | |
.
. 1 2 4 6 9 12
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.
. _ _ _ _ _ _ _ _
. _ _ _ _ _ _ _ _ |
. _ _ _ _ _|_ _ _|_ _ _|_ |
. _ _ | _ _ | _ _ | _ _ | |
. _ | | _ | | _ | | _ | | |
. | | | | | | | | | | | | |
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. 14 15 19 24
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. _ _ _ _ _ _ _ _ _ _
. _ _ _ _ _ _ _ _ _ _ _ _ |
. _ _ _ _ _ _ _|_ _ _ _|_ _ _ _|_ |
. _ _ | _ _ | _ _ | _ _ | |
. _ _|_ | _ _|_ | _ _|_ | _ _|_ | |
. _ _ | | _ _ | | _ _ | | _ _ | | |
. _ | | | _ | | | _ | | | _ | | | |
. | | | | | | | | | | | | | | | | |
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. 27 28 33 40
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Illustration of initial terms as vertices (or the number of steps from the origin) of a Dyck path:
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7 33
. /\
5 19 / \
. /\ / \
3 9 / \ 27 / \
2 4 /\ 14 / \ /\/ \
1 1 /\ / \ /\/ \ / 28 \
. /\/ \/ \/ 15 \/ \
. 0 2 6 12 24 40
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CROSSREFS
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Cf. A000041, A006128, A135010, A138137, A139250, A139582, A141285, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225610.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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