A284940 -id:A284940

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A285833

Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 107", based on the 5-celled von Neumann neighborhood.

+10
4

1, 11, 110, 1111, 1100, 111111, 11000, 11111111, 11110000, 1111111111, 1111100000, 111111111111, 111111000000, 11111111111111, 11111110000000, 1111111111111111, 1111111100000000, 111111111111111111, 111111111000000000, 11111111111111111111

(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, Apr 28 2017: (Start)

G.f.: (1 + 11*x - x^2 - 110*x^3 - 10000*x^4 + 10000*x^6 + 11000000*x^8 - 111000000*x^10 + 100000000*x^12) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)*(1 - 10*x^2)).

a(n) = 111*a(n-2) - 1110*a(n-4) + 1000*a(n-6) for n>6.

(End)

MATHEMATICA

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

code = 107; 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. A284940, A285834, A285835.

KEYWORD

nonn,easy

AUTHOR

Robert Price, Apr 27 2017

STATUS

approved

A285834

Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 107", based on the 5-celled von Neumann neighborhood.

+10
4

1, 3, 3, 15, 6, 63, 12, 255, 30, 1023, 62, 4095, 126, 16383, 254, 65535, 510, 262143, 1022, 1048575, 2046, 4194303, 4094, 16777215, 8190, 67108863, 16382, 268435455, 32766, 1073741823, 65534, 4294967295, 131070, 17179869183, 262142, 68719476735, 524286

(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, Apr 28 2017: (Start)

G.f.: (1+3*x-4*x^2-6*x^3-x^4+4*x^6+6*x^8-28*x^10+16*x^12) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-2*x^2)).

a(n) = 7*a(n-2) - 14*a(n-4) + 8*a(n-6) for n>6.

(End)

MATHEMATICA

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

code = 107; 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}]

LinearRecurrence[{0, 7, 0, -14, 0, 8}, {1, 3, 3, 15, 6, 63, 12, 255, 30, 1023, 62, 4095, 126}, 40] (* Harvey P. Dale, Feb 28 2023 *)

CROSSREFS

Cf. A284940, A285833, A285835.

KEYWORD

nonn,easy

AUTHOR

Robert Price, Apr 27 2017

STATUS

approved

A285835

Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 107", based on the 5-celled von Neumann neighborhood.

+10
4

1, 3, 6, 15, 12, 63, 24, 255, 240, 1023, 992, 4095, 4032, 16383, 16256, 65535, 65280, 262143, 261632, 1048575, 1047552, 4194303, 4192256, 16777215, 16773120, 67108863, 67100672, 268435455, 268419072, 1073741823, 1073709056, 4294967295, 4294901760

(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, Apr 28 2017: (Start)

G.f.: (1+3*x-x^2-6*x^3-16*x^4+16*x^6+192*x^8-448*x^10+256*x^12) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-2*x^2)).

a(n) = 2^n - 2^(n/2) for n>6 and even.

a(n) = 2^(n + 1) - 1 for n>6 and odd.

a(n) = 7*a(n-2) - 14*a(n-4) + 8*a(n-6) for n>6.

(End)

MATHEMATICA

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

code = 107; 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. A284940, A285833, A285834.

KEYWORD

nonn,easy

AUTHOR

Robert Price, Apr 27 2017

STATUS

approved

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