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A271004 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 245", based on the 5-celled von Neumann neighborhood. 4
1, 8, 4, 44, 13, 112, 25, 208, 37, 340, 49, 504, 61, 700, 73, 928, 85, 1188, 97, 1480, 109, 1804, 121, 2160, 133, 2548, 145, 2968, 157, 3420, 169, 3904, 181, 4420, 193, 4968, 205, 5548, 217, 6160, 229, 6804, 241, 7480, 253, 8188, 265, 8928, 277, 9700, 289 (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

Robert Price, Diagrams of the 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

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, Mar 28 2016: (Start)

a(n) = (-13-9*(-1)^n+4*(2+(-1)^n)*n-4*(-1+(-1)^n)*n^2)/2 for n>5.

a(n) = 6*n-11 for n>3 and even.

a(n) = 4*n^2+2*n-2 for n>5 and odd.

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>9.

G.f.: (1+8*x+x^2+20*x^3+4*x^4+4*x^5-3*x^6-4*x^7-3*x^8+8*x^9-4*x^11) / ((1-x)^3*(1+x)^3).

(End)

MATHEMATICA

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

code=245; 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}];

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

CROSSREFS

Sequence in context: A270677 A270901 A270934 * A271051 A046106 A112584

Adjacent sequences:  A271001 A271002 A271003 * A271005 A271006 A271007

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 28 2016

STATUS

approved

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Last modified July 6 14:10 EDT 2022. Contains 355110 sequences. (Running on oeis4.)