A284302 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 865", based on the 5-celled von Neumann neighborhood.
1, 0, 6, 10, 10, 46, 42, 238, 426, 750, 682, 2798, 2730, 10990, 10922, 44014, 43946, 178926, 174762, 781038, 764586, 3844846, 7252650, 11185134, 16427946, 48937710, 45787818, 178973422, 178956970, 715827950, 715827882, 2863311854, 2863311786, 11453250286
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
- Robert Price, Diagrams of first 20 stages
- 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
- Wolfram Research, Wolfram Atlas of Simple Programs
- Index entries for sequences related to cellular automata
- Index to 2D 5-Neighbor Cellular Automata
- Index to Elementary Cellular Automata
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 = 865; 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
Conjectures from Chai Wah Wu, May 04 2024: (Start)
a(n) = 4*a(n-2) + a(n-8) - 4*a(n-10) for n > 36.
G.f.: (4194304*x^36 + 16777216*x^35 + 19922944*x^34 - 4194304*x^33 + 12582912*x^32 + 4194304*x^31 - 4194304*x^30 - 786432*x^29 - 4259840*x^28 - 16842752*x^27 - 19922944*x^26 + 4194304*x^25 - 12582912*x^24 - 4194304*x^23 + 4194304*x^22 + 720896*x^21 + 65536*x^20 + 65536*x^19 + 3072*x^17 - 208*x^13 + 16*x^12 - 212*x^11 - 1024*x^10 - 202*x^9 + 257*x^8 + 54*x^7 + 2*x^6 + 6*x^5 - 14*x^4 + 10*x^3 + 2*x^2 + 1)/(4*x^10 - x^8 - 4*x^2 + 1). (End)
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