A270166 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 107", based on the 5-celled von Neumann neighborhood.

1, 6, 11, 52, 64, 173, 189, 386, 426, 747, 807, 1276, 1360, 2005, 2117, 2966, 3110, 4191, 4371, 5712, 5932, 7561, 7825, 9770, 10082, 12371, 12735, 15396, 15816, 18877, 19357, 22846, 23390, 27335, 27947, 32376, 33060, 38001, 38761, 44242, 45082, 51131, 52055

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 13 2016: (Start)

a(n) = 1/12*(-9*(23+(-1)^n)+(28-18*(-1)^n)*n-9*(-3+(-1)^n)*n^2+8*n^3) for n>6.

a(n) = (4*n^3+9*n^2+5*n-108)/6 for n>6 and even.

a(n) = (4*n^3+18*n^2+23*n-99)/6 for n>6 and odd.

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7) for n>13.

G.f.: (1+5*x+2*x^2+26*x^3+x^5-6*x^6-12*x^7+23*x^8+16*x^9-24*x^10-12*x^11+8*x^12+4*x^13) / ((1-x)^4*(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=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}];
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. A270164.

Sequence in context: A271279 A295498 A270158 * A270183 A270212 A077701

Adjacent sequences: A270163 A270164 A270165 * A270167 A270168 A270169

Keyword

nonn,easy

Author

Robert Price, Mar 12 2016

Status

approved