|
| |
|
|
A269876
|
|
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.
|
|
3
|
|
|
|
1, 5, 5, 37, 13, 97, 25, 185, 41, 301, 61, 445, 85, 617, 113, 817, 145, 1045, 181, 1301, 221, 1585, 265, 1897, 313, 2237, 365, 2605, 421, 3001, 481, 3425, 545, 3877, 613, 4357, 685, 4865, 761, 5401, 841, 5965, 925, 6557, 1013, 7177, 1105, 7825, 1201, 8501
(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
|
|
|
|
FORMULA
|
a(n) = (1+3*(-1)^n-2*(-3+(-1)^n)*n+(8-6*(-1)^n)*n^2)/4.
a(n) = (n^2+2*n+2)/2 for n even.
a(n) = (7*n^2+4*n-1)/2 for n odd.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>5.
G.f.: (1+5*x+2*x^2+22*x^3+x^4+x^5) / ((1-x)^3*(1+x)^3).
(End)
|
|
|
MATHEMATICA
|
code=43; stages=100;
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
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
