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A270683 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 190", based on the 5-celled von Neumann neighborhood. 2
1, 6, 18, 42, 74, 126, 186, 274, 370, 502, 642, 826, 1018, 1262, 1514, 1826, 2146, 2534, 2930, 3402, 3882, 4446, 5018, 5682, 6354, 7126, 7906, 8794, 9690, 10702, 11722, 12866, 14018, 15302, 16594, 18026, 19466, 21054, 22650, 24402, 26162, 28086, 30018, 32122 (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

FORMULA

Conjectures from Colin Barker, Mar 21 2016: (Start)

a(n) = (9+3*(-1)^n+7*n-3*(-1)^n*n+18*n^2+2*n^3)/6 for n>0.

a(n) = (n^3+9*n^2+2*n+6)/3 for n>0 and even.

a(n) = (n^3+9*n^2+5*n+3)/3 for n odd.

a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6) for n>6.

G.f.: (1+4*x+5*x^2+4*x^3-5*x^4-x^6) / ((1-x)^4*(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=190; 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. A270681.

Sequence in context: A286308 A015942 A009945 * A011930 A330844 A068293

Adjacent sequences:  A270680 A270681 A270682 * A270684 A270685 A270686

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 21 2016

STATUS

approved

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Last modified May 15 07:51 EDT 2021. Contains 343909 sequences. (Running on oeis4.)