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A182632 Toothpick sequence on the hexagonal net starting from a node. 14
0, 3, 9, 21, 33, 45, 69, 105, 129, 141, 165, 213, 273, 321, 381, 465, 513, 525, 549, 597, 657, 717, 801, 933, 1065, 1137, 1197, 1317, 1485, 1629, 1785, 1977, 2073, 2085, 2109, 2157, 2217, 2277, 2361, 2493, 2625, 2709, 2793, 2949, 3177, 3405, 3633 (list; graph; refs; listen; history; text; internal format)



A connected network of toothpicks is constructed by the following iterative procedure. At stage 1, place three toothpicks each of length 1 on a hexagonal net, as a propeller, joined at a node. At each subsequent stage, add two toothpicks (which could be called a single V-toothpick with a 120 degree corner) adjacent to each node which is the endpoint of a single toothpick.

The exposed endpoints of the toothpicks of the old generation are touched by the endpoints of the toothpicks of the new generation. In the graph, the edges of the hexagons become edges of the graph, and the graph grows such that the nodes that were 1-connected in the old generation are 3-connected in the new generation.

It turns out heuristically that this growth does not show frustration, ie., a free edge is never claimed by two adjacent exposed endpoints at the same stage; the rule of growing the network does apparently not need specifications to address such cases.

The sequence gives the number of toothpicks in the toothpick structure after n-th stage. A182633 (the first differences) gives the number of toothpicks added at n-th stage.

a(n) is also the number of components after n-th stage in a toothpick structure starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >=2. For example: consider that in A161644 a V-toothpick is also a polytoothpick with two components or toothpicks and a Y-toothpick is also a polytoothpick with three components or toothpicks. For more information about this comment see A161206, A160120 and A161644.

Has a behavior similar to A151723, A182840. - Omar E. Pol, Mar 07 2013


Table of n, a(n) for n=0..46.

David Applegate, The movie version

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to toothpick sequences

Index entries for sequences related to cellular automata


a(n) = 3*A182634(n).

a(n) = 1 + 2*A161644(n), n >= 1. - Omar E. Pol, Mar 07 2013


a(0)=0. At stage 1 we place 3 toothpicks connected to the initial grid point of the structure. Note that there are 3 exposed endpoints. At stage 2 we place 6 toothpicks, so a(2)=3+6=9, etc.


Cf. A139250, A160120, A161206, A161644, A182617, A182633, A182634, A182836, A182840.

Sequence in context: A029536 A331131 A089322 * A286590 A061978 A031886

Adjacent sequences:  A182629 A182630 A182631 * A182633 A182634 A182635




Omar E. Pol, Dec 07 2010



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Last modified July 24 14:45 EDT 2021. Contains 346273 sequences. (Running on oeis4.)