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A173456 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

%I A173456 #23 Feb 24 2021 02:48:19
%S A173456 0,1,9,21,25,53,89,93,121,157,169,253,361,365,393,429,441,525,633,645,
%T A173456 729,837,873,1125,1449,1453,1481,1517,1529,1613,1721,1733,1817,1925,
%U A173456 1961,2213,2537,2549,2633,2741,2777,3029,3353,3389,3641,3965,4073,4829,5801
%N A173456 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).
%C A173456 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 A173456 In order to construct this sequence we use the following rules:
%C A173456 - 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 A173456 - If n is congruent to 1 (mod 3), we turn "ON" the possible peninsula cells (For the definition of peninsula cell see A160117).
%C A173456 - If n is congruent to 2 (mod 3), we turn "ON" the cells around the cells turned "ON" at the generation n-1.
%C A173456 - Everything that is already ON remains ON.
%C A173456 A173457, the first differences, gives the number of cells turned "ON" at n-th stage.
%H A173456 Lars Blomberg, <a href="/A173456/b173456.txt">Table of n, a(n) for n = 0..5999</a>
%H A173456 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 A173456 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 A173456 a(0)=0, a(n)=a(n-1)+A173457(n), n>=1
%e A173456 Array begins:
%e A173456 0, 1, 9;
%e A173456 21, 25, 53;
%e A173456 89, 93, 121;
%e A173456 157, 169, 253;
%e A173456 361, 365, 393;
%e A173456 ...
%e A173456 If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
%e A173456 7...........7
%e A173456 .66.66.66.66.
%e A173456 .65556.65556.
%e A173456 ..545...545..
%e A173456 .65533.33556.
%e A173456 .66.32223.66.
%e A173456 .....212.....
%e A173456 .66.32223.66.
%e A173456 .65533.33556.
%e A173456 ..545...545..
%e A173456 .65556.65556.
%e A173456 .66.66.66.66.
%e A173456 7...........7
%Y A173456 Cf. A160117, A160118, A173457, A173458, A173459, A173460.
%K A173456 nonn
%O A173456 0,3
%A A173456 _Omar E. Pol_, Feb 18 2010
%E A173456 a(41)-a(48) from _Lars Blomberg_, Apr 22 2013