A273790 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 931", based on the 5-celled von Neumann neighborhood.
1, 6, 31, 80, 161, 282, 451, 676, 965, 1326, 1767, 2296, 2921, 3650, 4491, 5452, 6541, 7766, 9135, 10656, 12337, 14186, 16211, 18420, 20821, 23422, 26231, 29256, 32505, 35986, 39707, 43676, 47901, 52390, 57151, 62192, 67521, 73146, 79075, 85316, 91877, 98766
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..128
- 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.
Crossrefs
Cf. A273789.
Programs
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Mathematica
CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}]; code=931; 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}]; on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *) Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
Formula
Conjectures from Colin Barker, May 31 2016: (Start)
a(n) = (4*n^3+12*n^2+11*n-9)/3 for n>0.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: (1+2*x+13*x^2-12*x^3+4*x^4) / (1-x)^4. (End)
Conjectured e.g.f.: 4 + exp(x)*(4*x^3/3 + 8*x^2 + 9*x - 3). - Stefano Spezia, Dec 30 2024
Comments