OFFSET
0,3
COMMENTS
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.
LINKS
EXAMPLE
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 | | | | | | |
.
. 1 2 3 4 5 6 7
.
Illustration of initial terms:
.
. _ _ _ _ _ _
. _ _ _ _ _ _ _ _ |
. _ _ _ _ | _ | _ | |
. | | | | | | | | |
.
. 1 2 4 6 9 12
.
.
. _ _ _ _ _ _ _ _
. _ _ _ _ _ _ _ _ |
. _ _ _ _ _|_ _ _|_ _ _|_ |
. _ _ | _ _ | _ _ | _ _ | |
. _ | | _ | | _ | | _ | | |
. | | | | | | | | | | | | |
.
. 14 15 19 24
.
.
. _ _ _ _ _ _ _ _ _ _
. _ _ _ _ _ _ _ _ _ _ _ _ |
. _ _ _ _ _ _ _|_ _ _ _|_ _ _ _|_ |
. _ _ | _ _ | _ _ | _ _ | |
. _ _|_ | _ _|_ | _ _|_ | _ _|_ | |
. _ _ | | _ _ | | _ _ | | _ _ | | |
. _ | | | _ | | | _ | | | _ | | | |
. | | | | | | | | | | | | | | | | |
.
. 27 28 33 40
.
Illustration of initial terms as vertices (or the number of steps from the origin) of a Dyck path:
.
7 33
. /\
5 19 / \
. /\ / \
3 9 / \ 27 / \
2 4 /\ 14 / \ /\/ \
1 1 /\ / \ /\/ \ / 28 \
. /\/ \/ \/ 15 \/ \
. 0 2 6 12 24 40
.
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jul 28 2013
STATUS
approved