cp's OEIS Frontend

A160170 X-toothpick sequence on Z^3 lattice (see Comments for precise definition).

%I A160170 #33 Sep 22 2024 15:51:15
%S A160170 0,1,5,13,21,45,77,109,165,245,325,413,525,685,853,1093,1317,1661,
%T A160170 1981,2301,2645,3093,3621,4157,4861,5565,6437,7173,8053,8893,9917,
%U A160170 11005,12261,13589,14981,16397,17837,19341,20997
%N A160170 X-toothpick sequence on Z^3 lattice (see Comments for precise definition).
%C A160170 Here a "X-toothpick" is defined to be a cross with 4 endpoints and a midpoint. Also, a X-toothpick can be represented by set of four connected toothpicks forming a cross.
%C A160170 We start at stage 0 on the Z^3 lattice with no X-toothpicks.
%C A160170 At stage 1 place a X-toothpick.
%C A160170 Rule: each exposed endpoint of the X-toothpicks of the old generation must be touched by the midpoint of a X-toothpick of new generation (see illustrations).
%C A160170 The sequence gives the number of X-toothpicks in the three-dimensional structure after n-th stage. A170171 (the first differences) gives the number of X-toothpicks added at n-th stage.
%C A160170 For a similar sequence but starting with a single toothpick see A170876.
%C A160170 For the X-toothpick sequence on Z^2 lattice see A147562, the Ulam-Warburton cellular automaton.
%C A160170 For more information about the growth of toothpicks see A139250.
%H A160170 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.],
%H A160170 R. J. Mathar, <a href="/A160170/a160170.cpp.txt">C++ program</a>
%H A160170 R. J. Mathar, <a href="/A160170/a160170.pdf">Views after stages 1 to 10 (PDF)</a>
%H A160170 Olbaid Fractalium, <a href="https://www.youtube.com/watch?v=kc8mzwzRyHo">Toothpick sequence Part 1</a>, (Watch from minute 3:14), Youtube video (2023).
%H A160170 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 A160170 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%Y A160170 Cf. A139250, A160120, A160160, A160171, A170876.
%Y A160170 Cf. A147562. - Omar E. Pol, Mar 28 2011
%K A160170 nonn
%O A160170 0,3
%A A160170 _Omar E. Pol_, May 03 2009
%E A160170 C++ program, illustrations and more terms (a(6)-a(38)) based on Email from _R. J. Mathar_ dated Jan 10 2010.