A281216 -id:A281216

Search: a281216 -id:a281216

Displaying 1-3 of 3 results found.

page 1

A281214

Binary 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 342", based on the 5-celled von Neumann neighborhood.

+10
4

1, 11, 100, 1110, 10001, 111011, 1000100, 11101110, 100010001, 1110111011, 10001000100, 111011101110, 1000100010001, 11101110111011, 100010001000100, 1110111011101110, 10001000100010001, 111011101110111011, 1000100010001000100, 11101110111011101110

(list; graph; refs; listen; history; text; internal format)

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 Colin Barker, Jan 18 2017: (Start)

a(n) = 100*a(n-2) + a(n-4) - 100*a(n-6) for n>5.

G.f.: (1 + 11*x + 10*x^3) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)*(1 + x^2)).

(End)

MATHEMATICA

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

code = 342; 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]], 10], {i, 1, stages - 1}]

CROSSREFS

Cf. A281215, A281216, A281217.

KEYWORD

nonn,easy

AUTHOR

Robert Price, Jan 17 2017

STATUS

approved

A281215

Binary 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 342", based on the 5-celled von Neumann neighborhood.

+10
4

1, 11, 1, 111, 10001, 110111, 10001, 1110111, 100010001, 1101110111, 100010001, 11101110111, 1000100010001, 11011101110111, 1000100010001, 111011101110111, 10001000100010001, 110111011101110111, 10001000100010001, 1110111011101110111, 100010001000100010001

(list; graph; refs; listen; history; text; internal format)

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 Colin Barker, Jan 18 2017: (Start)

a(n) = a(n-2) + 10000*a(n-4) - 10000*a(n-6) for n>5.

G.f.: (1 + 11*x + 100*x^3) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)*(1 + 100*x^2)).

(End)

MATHEMATICA

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

code = 342; 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]], 10], {i, 1, stages - 1}]

CROSSREFS

Cf. A281214, A281216, A281217.

KEYWORD

nonn,easy

AUTHOR

Robert Price, Jan 17 2017

STATUS

approved

A281217

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

+10
4

1, 3, 1, 7, 17, 55, 17, 119, 273, 887, 273, 1911, 4369, 14199, 4369, 30583, 69905, 227191, 69905, 489335, 1118481, 3635063, 1118481, 7829367, 17895697, 58161015, 17895697, 125269879, 286331153, 930576247, 286331153, 2004318071, 4581298449, 14889219959

(list; graph; refs; listen; history; text; internal format)

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

Conjecture: a(n) = floor(16^(floor(n/4)+1)/15)*((n+1) mod 2) + 52*floor(16^(floor(n/4))/15)*0^((n+3) mod 4) + 7*floor(16^(floor(n/4)+1)/15)*0^((n+1) mod 4). - Karl V. Keller, Jr., Sep 27 2021

MATHEMATICA

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

code = 342; 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. A281214, A281215, A281216.

KEYWORD

nonn,easy

AUTHOR

Robert Price, Jan 17 2017

STATUS

approved

page 1

Search completed in 0.007 seconds