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A173460 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

%I A173460 #18 Feb 24 2021 02:48:19
%S A173460 0,1,9,21,29,81,93,105,189,225,253,441,453,465,549,585,621,873,909,
%T A173460 945,1197,1305,1397,2025,2037,2049,2133,2169,2205,2457,2493,2529,2781,
%U A173460 2889,2997,3753,3789,3825,4077,4185,4293,5049,5157,5265,6021,6345,6637,8649
%N A173460 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).
%C A173460 On the infinite square grid, we start at stage 0 with all cells in OFF state. At stage 1, we turn ON a single cell, in the central position.
%C A173460 In order to construct this sequence we use the following rules:
%C A173460 - If n is congruent to 0 (mod 3), we turn "ON" the cells around the vertex of every convex corner formed in the structure at the generation n-1. Note that every vertex is surrounded by three new "ON" cells.
%C A173460 - If n is congruent to 1 (mod 3), we turn "ON" the possible bridge cells and the possible peninsula cells (For the definition of bridge cell and of peninsula cell see A160117).
%C A173460 - If n is congruent to 2 (mod 3), we turn "ON" the cells around the cells turned "ON" at the generation n-1.
%C A173460 - Everything that is already ON remains ON.
%C A173460 A173461, the first differences, gives the number of cells turned "ON" at n-th stage.
%H A173460 Lars Blomberg, <a href="/A173460/b173460.txt">Table of n, a(n) for n = 0..6000</a>
%H A173460 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 A173460 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>
%F A173460 a(0)=0, a(n) = a(n-1) + A173461(n), n>=1. - [_Lars Blomberg_, Apr 23 2013]
%e A173460 Array begins:
%e A173460 0, 1, 9;
%e A173460 21, 29, 81;
%e A173460 93, 105, 189;
%e A173460 225, 253, 441;
%e A173460 453, 465, 549;
%e A173460 ...
%e A173460 If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
%e A173460 7..7.....7..7
%e A173460 .66.......66.
%e A173460 .65555555556.
%e A173460 7.545545545.7
%e A173460 ..553353355..
%e A173460 ..553222355..
%e A173460 ..545212545..
%e A173460 ..553222355..
%e A173460 ..553353355..
%e A173460 7.545545545.7
%e A173460 .65555555556.
%e A173460 .66.......66.
%e A173460 7..7.....7..7
%Y A173460 Cf. A160117, A160118, A173456, A173461, A173462, A173463.
%K A173460 nonn
%O A173460 0,3
%A A173460 _Omar E. Pol_, Feb 18 2010
%E A173460 a(18)-a(47) from _Lars Blomberg_, Apr 23 2013