|
| |
|
|
A269879
|
|
First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 43", based on the 5-celled von Neumann neighborhood.
|
|
1
|
|
|
|
4, 0, 32, -24, 84, -72, 160, -144, 260, -240, 384, -360, 532, -504, 704, -672, 900, -864, 1120, -1080, 1364, -1320, 1632, -1584, 1924, -1872, 2240, -2184, 2580, -2520, 2944, -2880, 3332, -3264, 3744, -3672, 4180, -4104, 4640, -4560, 5124, -5040, 5632, -5544
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
FORMULA
|
a(n) = (7+(-1)^n+8*(1+(-1)^n)*n+6*(-1)^n*n^2)/2.
a(n) = 3*n^2+8*n+4 for n even.
a(n) = -3*n^2+3 for n odd.
a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>4.
G.f.: 4*(1+x+6*x^2) / ((1-x)^2*(1+x)^3).
(End)
|
|
|
MATHEMATICA
|
code=43; 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
|
|
|
|
KEYWORD
|
sign,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
