A273420 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.

3, 17, 20, 31, 41, 47, 57, 63, 73, 79, 89, 95, 105, 111, 121, 127, 137, 143, 153, 159, 169, 175, 185, 191, 201, 207, 217, 223, 233, 239, 249, 255, 265, 271, 281, 287, 297, 303, 313, 319, 329, 335, 345, 351, 361, 367, 377, 383, 393, 399, 409, 415, 425, 431

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

Formula

Conjectures from Colin Barker, May 22 2016: (Start)

a(n) = 8+(-1)^n+8*n for n>2.

a(n) = 8*n+9 for n>2 and even.

a(n) = 8*n+7 for n>2 and odd.

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

G.f.: (3+14*x-3*x^3+7*x^4-5*x^5) / ((1-x)^2*(1+x)).

(End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=705; 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. A273417.

Sequence in context: A370106 A147845 A077778 * A022127 A273448 A298469

Adjacent sequences: A273417 A273418 A273419 * A273421 A273422 A273423

Keyword

nonn,easy

Author

Robert Price, May 22 2016

Status

approved