This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A225600 #56 Aug 22 2014 05:11:58 %S A225600 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 A225600 79,81,85,86,93,108,110,111,115,117,120,121,127,131,136,137,141,142, %U A225600 150,172,175,176,181,183,187,188,195,199,202,203,209,211,216,217,226,256 %N A225600 Toothpick sequence related to integer partitions (see Comments lines for definition). %C A225600 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 A225600 To define the sequence we use the following rules: %C A225600 We start in the first quadrant of the square grid with no toothpicks. %C A225600 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 A225600 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 A225600 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 A225600 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 A225600 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa408.jpg">Visualization of regions in a diagram for A006128</a> %H A225600 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 A225600 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a> %F A225600 a(A139582(n)) = a(2*A000041(n)) = 2*A006128(n) = A211978(n), n >= 1. %e A225600 For n = 30 the structure has 108 toothpicks, so a(30) = 108. %e A225600 . Diagram of regions %e A225600 Partitions of 7 and partitions of 7 %e A225600 . _ _ _ _ _ _ _ %e A225600 7 15 _ _ _ _ | %e A225600 4 + 3 _ _ _ _|_ | %e A225600 5 + 2 _ _ _ | | %e A225600 3 + 2 + 2 _ _ _|_ _|_ | %e A225600 6 + 1 11 _ _ _ | | %e A225600 3 + 3 + 1 _ _ _|_ | | %e A225600 4 + 2 + 1 _ _ | | | %e A225600 2 + 2 + 2 + 1 _ _|_ _|_ | | %e A225600 5 + 1 + 1 7 _ _ _ | | | %e A225600 3 + 2 + 1 + 1 _ _ _|_ | | | %e A225600 4 + 1 + 1 + 1 5 _ _ | | | | %e A225600 2 + 2 + 1 + 1 + 1 _ _|_ | | | | %e A225600 3 + 1 + 1 + 1 + 1 3 _ _ | | | | | %e A225600 2 + 1 + 1 + 1 + 1 + 1 2 _ | | | | | | %e A225600 1 + 1 + 1 + 1 + 1 + 1 + 1 1 | | | | | | | %e A225600 . %e A225600 . 1 2 3 4 5 6 7 %e A225600 . %e A225600 Illustration of initial terms: %e A225600 . %e A225600 . _ _ _ _ _ _ %e A225600 . _ _ _ _ _ _ _ _ | %e A225600 . _ _ _ _ | _ | _ | | %e A225600 . | | | | | | | | | %e A225600 . %e A225600 . 1 2 4 6 9 12 %e A225600 . %e A225600 . %e A225600 . _ _ _ _ _ _ _ _ %e A225600 . _ _ _ _ _ _ _ _ | %e A225600 . _ _ _ _ _|_ _ _|_ _ _|_ | %e A225600 . _ _ | _ _ | _ _ | _ _ | | %e A225600 . _ | | _ | | _ | | _ | | | %e A225600 . | | | | | | | | | | | | | %e A225600 . %e A225600 . 14 15 19 24 %e A225600 . %e A225600 . %e A225600 . _ _ _ _ _ _ _ _ _ _ %e A225600 . _ _ _ _ _ _ _ _ _ _ _ _ | %e A225600 . _ _ _ _ _ _ _|_ _ _ _|_ _ _ _|_ | %e A225600 . _ _ | _ _ | _ _ | _ _ | | %e A225600 . _ _|_ | _ _|_ | _ _|_ | _ _|_ | | %e A225600 . _ _ | | _ _ | | _ _ | | _ _ | | | %e A225600 . _ | | | _ | | | _ | | | _ | | | | %e A225600 . | | | | | | | | | | | | | | | | | %e A225600 . %e A225600 . 27 28 33 40 %e A225600 . %e A225600 Illustration of initial terms as vertices (or the number of steps from the origin) of a Dyck path: %e A225600 . %e A225600 7 33 %e A225600 . /\ %e A225600 5 19 / \ %e A225600 . /\ / \ %e A225600 3 9 / \ 27 / \ %e A225600 2 4 /\ 14 / \ /\/ \ %e A225600 1 1 /\ / \ /\/ \ / 28 \ %e A225600 . /\/ \/ \/ 15 \/ \ %e A225600 . 0 2 6 12 24 40 %e A225600 . %Y A225600 Cf. A000041, A006128, A135010, A138137, A139250, A139582, A141285, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225610. %K A225600 nonn %O A225600 0,3 %A A225600 _Omar E. Pol_, Jul 28 2013