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A161645 First differences of A161644: number of new ON cells at generation n of the triangular cellular automaton described in A161644.

%I A161645 #43 Mar 11 2021 23:07:52
%S A161645 0,1,3,6,6,6,12,18,12,6,12,24,30,24,30,42,24,6,12,24,30,30,42,66,66,
%T A161645 36,30,60,84,72,78,96,48,6,12,24,30,30,42,66,66,42,42,78,114,114,114,
%U A161645 150,138,60,30,60,84,90,114,174,198,132,90,144,210,192,192,210,96,6,12,24
%N A161645 First differences of A161644: number of new ON cells at generation n of the triangular cellular automaton described in A161644.
%C A161645 See the comments in A161644.
%C A161645 It appears that a(n) is also the number of V-toothpicks or Y-toothpicks added at the n-th stage in a toothpick structure on hexagonal net, starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >=2 (see A161206, A160120, A182633). - _Omar E. Pol_, Dec 07 2010
%D A161645 R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Describes the dual structure where new triangles are joined at vertices rather than edges.]
%H A161645 Rémy Sigrist, <a href="/A161645/b161645.txt">Table of n, a(n) for n = 0..10000</a>
%H A161645 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A161645 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 A161645 R. Reed, <a href="/A005448/a005448_1.pdf">The Lemming Simulation Problem</a>, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
%H A161645 N. J. A. Sloane, <a href="/A161644/a161644_1.png">Illustration of first 7 generations of A161644 and A295560 (edge-to-edge version)</a>
%H A161645 N. J. A. Sloane, <a href="/A161644/a161644_2.png">Illustration of first 11 generations of A161644 and A295560 (vertex-to-vertex version)</a> [Include the 6 cells marked x to get A161644(11), exclude them to get A295560(11).]
%H A161645 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>
%e A161645 From _Omar E. Pol_, Apr 08 2015: (Start)
%e A161645 The positive terms written as an irregular triangle in which the row lengths are the terms of A011782:
%e A161645 1;
%e A161645 3;
%e A161645 6,6;
%e A161645 6,12,18,12;
%e A161645 6,12,24,30,24,30,42,24;
%e A161645 6,12,24,30,30,42,66,66,36,30,60,84,72,78,96,48;
%e A161645 6,12,24,30,30,42,66,66,42,42,78,114,114,114,150,138,60,30,60,84,90,114,174,198,132,90,144,210,192,192,210,96;
%e A161645 ...
%e A161645 It appears that the right border gives A003945.
%e A161645 (End)
%Y A161645 Cf. A161644, A139251, A160161, A161207, A182633, A295559, A295560.
%Y A161645 See A342271 for a(n)/3.
%K A161645 nonn,look
%O A161645 0,3
%A A161645 _David Applegate_ and _N. J. A. Sloane_, Jun 15 2009