A273911 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 614", based on the 5-celled von Neumann neighborhood.
1, 3, 5, 11, 17, 55, 69, 219, 257, 775, 1301, 2923, 4113, 12407, 20805, 46811, 65537, 196615, 327701, 721003, 1114385, 3606391, 4527173, 14380507, 16777473, 50333447, 83891477, 184576875, 285282321, 923234423, 1158959429, 3681400539, 4294967297, 12884901895
Offset: 0
References
- S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
Links
- Robert Price, Table of n, a(n) for n = 0..126
- N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- S. Wolfram, A New Kind of Science
- Index entries for sequences related to cellular automata
- Index to 2D 5-Neighbor Cellular Automata
- Index to Elementary Cellular Automata
Programs
-
Mathematica
CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}]; code=614; stages=128; rule=IntegerDigits[code,2,10]; g=2*stages+1; (* Maximum size of grid *) a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *) ca=a; ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}]; PrependTo[ca,a]; (* Trim full grid to reflect growth by one cell at each stage *) k=(Length[ca[[1]]]+1)/2; ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}]; Table[FromDigits[Part[ca[[i]][[i]],Range[i,2*i-1]],2], {i,1,stages-1}]
Formula
a(n) = Sum_{k=0..n} 2^k*(A383609(n, k) mod 2). - Mélika Tebni, May 16 2025
Comments