A282987 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 541", based on the 5-celled von Neumann neighborhood.

1, 1, 1, 9, 9, 9, 105, 137, 105, 777, 1513, 1929, 1129, 9993, 9705, 10121, 103529, 153353, 67049, 911241, 1119337, 644873, 6620649, 8710025, 6558825, 53729033, 94045673, 80275337, 107746409, 858248969, 1505166825, 1284827017, 1725174889, 13728470793

Offset

0,4

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 06 2024: (Start)

a(n) = 17*a(n-4) - 16*a(n-8) for n > 33.

G.f.: (-2097152*x^33 - 1572864*x^32 + 59506688*x^31 + 12320768*x^30 - 44826624*x^29 + 14155776*x^28 - 53215232*x^27 - 17432576*x^26 + 45219840*x^25 - 10813440*x^24 - 6619136*x^23 + 5636096*x^22 - 1802240*x^21 - 622592*x^20 + 770048*x^19 - 73728*x^18 - 4096*x^17 + 86016*x^16 - 20480*x^15 - 14336*x^14 - 3072*x^13 - 512*x^12 - 256*x^11 - 256*x^10 + 640*x^9 - 32*x^8 - 16*x^7 + 88*x^6 - 8*x^5 - 8*x^4 + 9*x^3 + x^2 + x + 1)/(16*x^8 - 17*x^4 + 1). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 541; 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. A282984, A282985, A282986.

Sequence in context: A282910 A282599 A282957 * A118662 A305503 A175219

Adjacent sequences: A282984 A282985 A282986 * A282988 A282989 A282990

Keyword

nonn,easy

Author

Robert Price, Feb 26 2017

Status

approved