A284026 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 782", based on the 5-celled von Neumann neighborhood.

1, 3, 7, 13, 25, 59, 127, 215, 407, 957, 2029, 3429, 6437, 15245, 32665, 55067, 103967, 245271, 518935, 876821, 1646865, 3899667, 8355843, 14090241, 26607617, 62783491, 132841475, 224460801, 421593089, 998309891, 2139095043, 3607101441, 6811549697

Offset

0,2

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) = a(n-1) - a(n-2) + a(n-3) + 256*a(n-8) - 256*a(n-9) + 256*a(n-10) - 256*a(n-11) for n > 32.

G.f.: (4096*x^32 + 5120*x^30 + 1024*x^28 + 2560*x^27 - 1024*x^26 + 1536*x^25 - 4368*x^24 - 2560*x^23 - 3348*x^22 - 2560*x^21 - 772*x^20 - 10*x^19 - 1020*x^18 + 378*x^17 - 240*x^16 - 22*x^15 - 52*x^14 - 32*x^13 - 16*x^12 - 98*x^11 - 16*x^10 + 126*x^9 + 4*x^8 + 122*x^7 + 80*x^6 + 40*x^5 + 16*x^4 + 8*x^3 + 5*x^2 + 2*x + 1)/(256*x^11 - 256*x^10 + 256*x^9 - 256*x^8 - x^3 + x^2 - x + 1). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 782; 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[1, i]], 2], {i, 1, stages - 1}]

Crossrefs

Cf. A284024, A284025, A284027.

Sequence in context: A376448 A363143 A282913 * A099810 A283177 A284297

Adjacent sequences: A284023 A284024 A284025 * A284027 A284028 A284029

Keyword

nonn,easy

Author

Robert Price, Mar 18 2017

Status

approved