

A182840


Toothpick sequence on hexagonal net.


17



0, 1, 5, 13, 27, 43, 57, 81, 119, 151, 165, 189, 235, 299, 353, 409, 495, 559, 573, 597, 643, 707, 769, 849, 975, 1119, 1205, 1261, 1371, 1539, 1697, 1841, 2039, 2167, 2181, 2205, 2251, 2315, 2377, 2457, 2583, 2727, 2821, 2901, 3043, 3267, 3505, 3729, 4015
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

Rules:
 Each new toothpick must lie on the hexagonal net such that the toothpick endpoints coincide with two consecutive nodes.
 Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of two toothpicks of new generation.
The sequence gives the number of toothpicks after n stages. A182841 (the first differences) gives the number added at the nth stage.
The toothpick structure has polygons in which there are uncovered grid points, the same as A160120 and A161206. For more information see A139250.
Has a behavior similar to A151723, A182632.  Omar E. Pol, Feb 28 2013


LINKS

Olaf Voß, Table of n, a(n) for n = 0..1000
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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Olaf Voß, Illustration of initial terms
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata


EXAMPLE

We start at stage 0 with no toothpicks.
At stage 1 we place a toothpick anywhere in the plane (For example, in vertical position). There are two exposed endpoints, so a(1)=1.
At stage 2 we place 4 toothpicks. Two new toothpicks touching each exposed endpoint. So a(2)=1+4=5. There are 4 exposed endpoints.
At stage 3 we place 8 toothpicks. a(3)=5+8=13. The structure has 8 exposed endpoints.
At stage 4 we place 14 toothpicks (Not 16) because there are 4 endpoints that are touched by new 8 toothpicks but there are 4 endpoints that are touched by only 6 new toothpicks (not 8), so a(4)=13+14=27.
After 4 stages the toothpick structure has 4 hexagons and 8 exposed endpoints.


CROSSREFS

Cf. A139250, A160120, A161206, A182632, A182634.
Sequence in context: A256111 A322417 A160420 * A301675 A147411 A008580
Adjacent sequences: A182837 A182838 A182839 * A182841 A182842 A182843


KEYWORD

nonn


AUTHOR

Omar E. Pol, Dec 09 2010


EXTENSIONS

More terms from Olaf Voß, Dec 24 2010
Wiki link added by Olaf Voß, Jan 14 2011


STATUS

approved



