A264768 - OEIS

%I #19 Jul 24 2024 14:46:18

%S 0,0,0,0,4,4,4,4,28,28,28,28,44,44,44,44,140,140,140,140,156,156,156,

%T 156,236,236,236,236,284,284,284,284,620,620,620,620,636,636,636,636,

%U 716,716,716,716,764,764,764,764,1068,1068,1068,1068,1116,1116,1116

%N Number of "SURROUNDED" cells after n stages 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 "SURROUNDED" cell is one in which all neighbors have been turned ON but itself is not 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 SURROUNDED.

%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 Number of SURROUNDED 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 JungHwan Min, <a href="/A264768/b264768.txt">Table of n, a(n) for n = 0..2500</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>

%t A264768[0] = 0; A264768[n_] := Count[CellularAutomaton[{87187427555196485724117344277761165884880477484970720900802366279596402859298341814252922040475078175743656473048023, 3, {{0, 1}, {-1, 0}, {0, 0}, {1, 0}, {0, -1}}}, {{{1}}, 0}, {{{n}}}], 2, 2] (* _JungHwan Min_, Jan 25 2016 *)

%t A264768[0] = 0; A264768[n_] := Total[With[{m = n - 1}, CellularAutomaton[{256, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, #, {{{1}}}] &@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, {{{m}}}]], 2] (* _JungHwan Min_, Jan 25 2016 *)

%Y Cf. A147562, A147582, A260490, A264039.

%Y Cumulative sums of A264769.

%K nonn

%O 0,5

%A _Robert Price_, Nov 24 2015