A270946 -id:A270946

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A270948

Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 229", based on the 5-celled von Neumann neighborhood.

+10
1

1, 9, 13, 53, 70, 174, 194, 383, 435, 759, 831, 1296, 1392, 2040, 2137, 2949, 3153, 4254, 4418, 5770, 6042, 7674, 7938, 9891, 10271, 12647, 13120, 15828, 16304, 19413, 19941, 23481, 24158, 28218, 28991, 33439, 34247, 39300, 40188, 45792, 46765, 53013, 54170

(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..128

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

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

MATHEMATICA

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

code=229; 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}];

on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *)

CROSSREFS

Cf. A270946.

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 26 2016

STATUS

approved

A270949

First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 229", based on the 5-celled von Neumann neighborhood.

+10
1

7, -4, 36, -23, 87, -84, 169, -137, 272, -252, 393, -369, 552, -551, 715, -608, 897, -937, 1188, -1080, 1360, -1368, 1689, -1573, 1996, -1903, 2235, -2232, 2633, -2581, 3012, -2863, 3383, -3287, 3675, -3640, 4245, -4165, 4716, -4631, 5275, -5091, 5683, -5592

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

OFFSET

0,1

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..127

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

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

MATHEMATICA

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

code=229; 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}];

on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

Table[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *)

CROSSREFS

Cf. A270946.

KEYWORD

sign,easy

AUTHOR

Robert Price, Mar 26 2016

STATUS

approved

A270947

Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 229", based on the 5-celled von Neumann neighborhood.

+10
0

1, 8, 40, 189, 812, 3540, 14224, 58672, 235520, 943648, 3785836, 15139112, 60577080, 242359424, 969552509, 3878246660

(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

Table of n, a(n) for n=0..15.

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

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

MATHEMATICA

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

code=229; 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}];

on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *)

CROSSREFS

Cf. A270946.

KEYWORD

nonn,more

AUTHOR

Robert Price, Mar 26 2016

EXTENSIONS

a(8)-a(15) from Lars Blomberg, Jun 07 2016

STATUS

approved

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