|
|
A278900
|
|
Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 107", based on the 5-celled von Neumann neighborhood.
|
|
4
|
|
|
1, 1, 1, 15, 4, 55, 0, 247, 20, 983, 20, 4087, 84, 16215, 84, 65495, 340, 261463, 340, 1048407, 1364, 4191575, 1364, 16776535, 5460, 67097943, 5460, 268432727, 21844, 1073698135, 21844, 4294956375, 87380, 17179694423, 87380, 68719433047, 349524, 274877207895
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
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) = 5*a(n-2) - 20*a(n-6) + 16*a(n-8) for n>15.
G.f.: (1 +x -4*x^2 +10*x^3 -x^4 -20*x^5 -8*x^7 +24*x^8 +32*x^9 -16*x^10 +32*x^11 -80*x^12 -160*x^13 +64*x^14 +128*x^15) / ((1 -x)*(1 +x)*(1 -2*x)*(1 +2*x)*(1 -2*x^2)*(1 +2*x^2)).
(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, k}];
Table[FromDigits[Part[ca[[i]][[i]], Range[1, i]], 2], {i, 1, stages - 1}]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|