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A260490
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Number of cells that are "POISONED" at the n-th stage of the "Ulam-Warburton" two-dimensional cellular automaton.
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3
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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, 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, 264, 0, 72, 0, 288, 0, 216, 0, 512, 0, 8, 0, 32, 0, 24, 0, 88, 0
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OFFSET
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0,3
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COMMENTS
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Previous analysis of this cellular automaton have concentrated on cells that turn "ON". This sequence examines those cells that are never turned ON.
This cellular automaton is generated by Rule 686 using the Wolfram numbering scheme.
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.
See A147562 for extensive definitions, references and links for this cellular automaton.
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.
Observations:
Cells are referenced by their coordinates on the x,y-plane with the initial cell at (0,0).
G(i,j) is the generation where cell (i,j) is turned ON.
P(i,j) is the generation where cell (i,j) is POISONED.
Due to symmetry, analysis of only the (+,+) quadrant is necessary.
G(0,j) = j+1;
G(i,0) = i+1;
G(k,2^n-1-k) = 2^n;
G(2^n-1-k,k) = 2^n;
G(1,j) = j+2, when j is even;
G(i,1) = i+2, when i is even;
P(1,j) = j+1, when j is odd;
P(i,1) = i+1, when i is odd;
P(i,j) = k, when i,j are odd (a formula for k is not known at this time);
P(i,j) = 2^k when i=j>0, k=floor(log_2(i-1))+2.
After iterations 2^k, all cells with i+j<=2^k are either ON or POISONED.
On iterations 2^k+1, only 4 cells turned on: (0,2^k), (2^k,0), (0,-2^k), (-2^k,0).
Newly turned ON cells are always adjacent to one turned ON in the previous generation.
"POISONING" only occurs at even number (>0) stages.
Number of POISONED cells approach 1/2 the number of ON cells as n increases.
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REFERENCES
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D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.
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.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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