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%I #56 Aug 22 2014 05:11:58
%S 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,
%T 79,81,85,86,93,108,110,111,115,117,120,121,127,131,136,137,141,142,
%U 150,172,175,176,181,183,187,188,195,199,202,203,209,211,216,217,226,256
%N Toothpick sequence related to integer partitions (see Comments lines for definition).
%C 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.
%C To define the sequence we use the following rules:
%C We start in the first quadrant of the square grid with no toothpicks.
%C 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).
%C 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).
%C 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.
%C 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.
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa408.jpg">Visualization of regions in a diagram for A006128</a>
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F a(A139582(n)) = a(2*A000041(n)) = 2*A006128(n) = A211978(n), n >= 1.
%e For n = 30 the structure has 108 toothpicks, so a(30) = 108.
%e . Diagram of regions
%e Partitions of 7 and partitions of 7
%e . _ _ _ _ _ _ _
%e 7 15 _ _ _ _ |
%e 4 + 3 _ _ _ _|_ |
%e 5 + 2 _ _ _ | |
%e 3 + 2 + 2 _ _ _|_ _|_ |
%e 6 + 1 11 _ _ _ | |
%e 3 + 3 + 1 _ _ _|_ | |
%e 4 + 2 + 1 _ _ | | |
%e 2 + 2 + 2 + 1 _ _|_ _|_ | |
%e 5 + 1 + 1 7 _ _ _ | | |
%e 3 + 2 + 1 + 1 _ _ _|_ | | |
%e 4 + 1 + 1 + 1 5 _ _ | | | |
%e 2 + 2 + 1 + 1 + 1 _ _|_ | | | |
%e 3 + 1 + 1 + 1 + 1 3 _ _ | | | | |
%e 2 + 1 + 1 + 1 + 1 + 1 2 _ | | | | | |
%e 1 + 1 + 1 + 1 + 1 + 1 + 1 1 | | | | | | |
%e .
%e . 1 2 3 4 5 6 7
%e .
%e Illustration of initial terms:
%e .
%e . _ _ _ _ _ _
%e . _ _ _ _ _ _ _ _ |
%e . _ _ _ _ | _ | _ | |
%e . | | | | | | | | |
%e .
%e . 1 2 4 6 9 12
%e .
%e .
%e . _ _ _ _ _ _ _ _
%e . _ _ _ _ _ _ _ _ |
%e . _ _ _ _ _|_ _ _|_ _ _|_ |
%e . _ _ | _ _ | _ _ | _ _ | |
%e . _ | | _ | | _ | | _ | | |
%e . | | | | | | | | | | | | |
%e .
%e . 14 15 19 24
%e .
%e .
%e . _ _ _ _ _ _ _ _ _ _
%e . _ _ _ _ _ _ _ _ _ _ _ _ |
%e . _ _ _ _ _ _ _|_ _ _ _|_ _ _ _|_ |
%e . _ _ | _ _ | _ _ | _ _ | |
%e . _ _|_ | _ _|_ | _ _|_ | _ _|_ | |
%e . _ _ | | _ _ | | _ _ | | _ _ | | |
%e . _ | | | _ | | | _ | | | _ | | | |
%e . | | | | | | | | | | | | | | | | |
%e .
%e . 27 28 33 40
%e .
%e Illustration of initial terms as vertices (or the number of steps from the origin) of a Dyck path:
%e .
%e 7 33
%e . /\
%e 5 19 / \
%e . /\ / \
%e 3 9 / \ 27 / \
%e 2 4 /\ 14 / \ /\/ \
%e 1 1 /\ / \ /\/ \ / 28 \
%e . /\/ \/ \/ 15 \/ \
%e . 0 2 6 12 24 40
%e .
%Y Cf. A000041, A006128, A135010, A138137, A139250, A139582, A141285, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225610.
%K nonn
%O 0,3
%A _Omar E. Pol_, Jul 28 2013
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