A160239 Number of "ON" cells in a 2-dimensional cellular automaton ("Fredkin's Replicator") evolving according to the rule that a cell is ON in a given generation if and only if there was an odd number of ON cells among the eight nearest neighbors in the preceding generation, starting with one ON cell.
1, 8, 8, 24, 8, 64, 24, 112, 8, 64, 64, 192, 24, 192, 112, 416, 8, 64, 64, 192, 64, 512, 192, 896, 24, 192, 192, 576, 112, 896, 416, 1728, 8, 64, 64, 192, 64, 512, 192, 896, 64, 512, 512, 1536, 192, 1536, 896, 3328, 24, 192, 192, 576, 192, 1536, 576, 2688, 112, 896, 896, 2688, 416, 3328, 1728, 6784
Offset: 0
Examples
From _Omar E. Pol_, Jul 22 2014 (Start): Written as an irregular triangle in which row lengths is A011782 the sequence begins: 1; 8; 8, 24; 8, 64, 24, 112; 8, 64, 64, 192, 24, 192, 112, 416; 8, 64, 64, 192, 64, 512, 192, 896, 24, 192, 192, 576, 112, 896, 416, 1728; 8, 64, 64, 192, 64, 512, 192, 896, 64, 512, 512, 1536, 192, 1536, 896, 3328, 24, 192, 192, 576, 192, 1536, 576, 2688, 112, 896, 896, 2688, 416, 3328, 1728, 6784; (End) Right border gives A246030. - _Omar E. Pol_, Jan 29 2015 [This is simply a restatement of the theorem that this sequence is the Run Length Transform of A246030. - _N. J. A. Sloane_, Jan 29 2015] . From _Omar E. Pol_, Mar 18 2015 (Start): Also, the sequence can be written as an irregular tetrahedron as shown below: 1; .. 8; .. 8; 24; ......... 8, 64; 24; 112; ................... 8, 64, 64, 192; 24, 192; 112; 416; ..................................... 8, 64, 64, 192, 64, 512,192, 896; 24, 192, 192, 576; 112, 896; 416; 1728; ....................................................................... 8, 64, 64, 192, 64, 512,192, 896,64,512,512,1536,192,1536,896,3328; 24, 192, 192, 576,192,1536,576,2688; 112, 896, 896,2688; 416,3328; 1728; 6784; ... Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k). On the other hand, it appears that the configuration of ON cells of T(s,r,k) is also the central part of the configuration of ON cells of T(s+1,r+1,k). (End)
Links
- N. J. A. Sloane, Table of n, a(n) for n = 0..10000
- Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
- Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
- Charles R Greathouse IV, Compact illustration of generations 0..17
- N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
- N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
- N. J. A. Sloane, Enlarged illustration of first 16 generations (pdf).
- N. J. A. Sloane, Illustration for a(15) = 416 (png).
- N. J. A. Sloane, Illustration for a(15) = 416 (pdf).
- N. J. A. Sloane, Illustration for a(31) = 1728.
- N. J. A. Sloane, Illustration for a(63) = 6784.
- N. J. A. Sloane, Illustration for a(127) = 27392 (tiff).
- N. J. A. Sloane, Illustration for a(127) = 27392 (png).
- N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
- Alexander Yu. Vlasov, Modelling reliability of reversible circuits with 2D second-order cellular automata, arXiv:2312.13034 [nlin.CG], 2023. See page 13.
- Index entries for sequences related to cellular automata
Crossrefs
Programs
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Haskell
import Data.List (transpose) a160239 n = a160239_list !! n a160239_list = 1 : (concat $ transpose [a8, hs, zipWith (+) (map (* 2) hs) a8, tail a160239_list]) where a8 = map (* 8) a160239_list; hs = h a160239_list; h (_:x:xs) = x : h xs -- Reinhard Zumkeller, Feb 13 2015 -
Maple
# From N. J. A. Sloane, Jan 19 2015: f:=proc(n) option remember; if n=0 then RETURN(1); elif n mod 2 = 0 then RETURN(f(n/2)) elif n mod 4 = 1 then RETURN(8*f((n-1)/4)) else RETURN(f(n-2)+2*f((n-1)/2)); fi; end; [seq(f(n),n=0..255)];
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Mathematica
A160239[n_] := CellularAutomaton[{52428, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{n}}][[1]] // Total@*Total (* Charles R Greathouse IV, Aug 21 2014 *) ArrayPlot /@ CellularAutomaton[{52428, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, 30] (* Charles R Greathouse IV, Aug 21 2014 *)
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PARI
A160239=[];a(n)={if(n>#A160239,A160239=concat(A160239,vector(n-#A160239)),n||return(1);A160239[n]&&return(A160239[n]));A160239[n]=if(bittest(n,0),if(bittest(n,1),a(n-2)+2*a(n\2),a(n\4)*8),a(n\2))} \\ M. F. Hasler, May 10 2016
Formula
a(0) = 1; a(2t)=a(t), a(4t+1)=8*a(t), a(4t+3)=2*a(2t+1)+8*a(t) for t >= 0. (Conjectured by Hrothgar, Jul 11 2014; proved by N. J. A. Sloane, Oct 04 2014.)
For n >= 2, a(n) = 8^r * Product_{lengths i of runs of 1 in binary expansion of n} R(i), where r is the number of runs of 1 in the binary expansion of n and R(i) = A083424(i-1) = (5*4^(i-1)+(-2)^(i-1))/6. Note that row i of the table in A245562 lists the lengths of runs of 1 in binary expansion of i. Example: n=7 = 111 in binary, so r=1, i=3, R(3) = A083424(2) = 14, and so a(7) = 8^1*14 = 112. That is, this sequence is the Run Length Transform of A246030. - N. J. A. Sloane, Oct 04 2014
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product). - N. J. A. Sloane, Aug 25 2014
Extensions
Offset changed to 1 by Hrothgar, Jul 11 2014
Offset reverted to 0 by N. J. A. Sloane, Jan 19 2015
Comments