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%I #19 Jun 27 2021 03:39:53
%S 0,0,4,0,12,0,8,0,32,0,8,0,32,0,24,0,80,0,8,0,32,0,24,0,88,0,24,0,96,
%T 0,72,0,200,0,8,0,32,0,24,0,88,0,24,0,96,0,72,0,224,0,24,0,96,0,72,0,
%U 264,0,72,0,288,0,216,0,512,0,8,0,32,0,24,0,88,0
%N Number of cells that are "POISONED" at the n-th stage of the "Ulam-Warburton" two-dimensional cellular automaton.
%C Previous analysis of this cellular automaton have concentrated on cells that turn "ON". This sequence examines those cells that are never turned ON.
%C This cellular automaton is generated by Rule 686 using the Wolfram numbering scheme.
%C A "POISONED" cell is one in which no further generation can possibly utilize that cell. That is, previous generations have ensured that more than one neighbor has been turned ON.
%C See A147562 for extensive definitions, references and links for this cellular automaton.
%C Note that the offset is zero, which implies that the initial cell is at stage n=1. This corresponds to that of A147562 where a(0)=0, a(1)=1, a(2)=5, etc. The Singmaster reference implies a(0)=1, a(1)=5, etc. The choice of offset is arbitrary and neither seems to be ideal.
%C Observations:
%C Cells are referenced by their coordinates on the x,y-plane with the initial cell at (0,0).
%C G(i,j) is the generation where cell (i,j) is turned ON.
%C P(i,j) is the generation where cell (i,j) is POISONED.
%C Due to symmetry, analysis of only the (+,+) quadrant is necessary.
%C G(0,j) = j+1;
%C G(i,0) = i+1;
%C G(k,2^n-1-k) = 2^n;
%C G(2^n-1-k,k) = 2^n;
%C G(1,j) = j+2, when j is even;
%C G(i,1) = i+2, when i is even;
%C P(1,j) = j+1, when j is odd;
%C P(i,1) = i+1, when i is odd;
%C P(i,j) = k, when i,j are odd (a formula for k is not known at this time);
%C P(i,j) = 2^k when i=j>0, k=floor(log_2(i-1))+2.
%C After iterations 2^k, all cells with i+j<=2^k are either ON or POISONED.
%C On iterations 2^k+1, only 4 cells turned on: (0,2^k), (2^k,0), (0,-2^k), (-2^k,0).
%C Newly turned ON cells are always adjacent to one turned ON in the previous generation.
%C "POISONING" only occurs at even number (>0) stages.
%C Number of POISONED cells approach 1/2 the number of ON cells as n increases.
%D D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.
%D S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.
%H Robert Price, <a href="/A260490/b260490.txt">Table of n, a(n) for n = 0..1030</a>
%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.
%H 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 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%Y Cf. A147562, A147582, A264039.
%K nonn
%O 0,3
%A _Robert Price_, Nov 10 2015
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