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

0,3

Comments

Initialized with a single black (ON) cell at stage zero.

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

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)

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}]

Crossrefs

Cf. A284299, A284300, A284301.

Sequence in context: A315122 A087873 A284238 * A284350 A333410 A377421

Adjacent sequences: A284299 A284300 A284301 * A284303 A284304 A284305

Keyword

nonn,easy

Author

Robert Price, Mar 24 2017

Status

approved