A264769 Number of cells that are "SURROUNDED" at the n-th stage of the "Ulam-Warburton" two-dimensional cellular automaton.

0, 0, 0, 0, 4, 0, 0, 0, 24, 0, 0, 0, 16, 0, 0, 0, 96, 0, 0, 0, 16, 0, 0, 0, 80, 0, 0, 0, 48, 0, 0, 0, 336, 0, 0, 0, 16, 0, 0, 0, 80, 0, 0, 0, 48, 0, 0, 0, 304, 0, 0, 0, 48, 0, 0, 0, 240, 0, 0, 0, 144, 0, 0, 0, 1104, 0, 0, 0, 16, 0, 0, 0, 80, 0, 0, 0, 48, 0, 0

Offset

0,5

Comments

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 "SURROUNDED" cell is one in which all neighbors have been turned ON but itself is not 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 SURROUNDED.

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.

Number of SURROUNDED cells approach 1/2 the number of ON cells as n increases.

References

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.

Links

Robert Price, Table of n, a(n) for n = 0..10000

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to cellular automata

Crossrefs

Cf. A147562, A147582, A260490, A264039.

Sequence in context: A298825 A265831 A348732 * A169765 A169768 A057386

Adjacent sequences: A264766 A264767 A264768 * A264770 A264771 A264772

Keyword

nonn

Author

Robert Price, Nov 24 2015

Status

approved