login
A271093
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 275", based on the 5-celled von Neumann neighborhood.
1
1, 6, 11, 51, 56, 168, 173, 389, 394, 746, 751, 1271, 1276, 1996, 2001, 2953, 2958, 4174, 4179, 5691, 5696, 7536, 7541, 9741, 9746, 12338, 12343, 15359, 15364, 18836, 18841, 22801, 22806, 27286, 27291, 32323, 32328, 37944, 37949, 44181, 44186, 51066, 51071
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.
FORMULA
Conjectures from Colin Barker, Mar 30 2016: (Start)
a(n) = (39+33*(-1)^n-2*(1+12*(-1)^n)*n-12*(-2+(-1)^n)*n^2+8*n^3)/12 for n>0.
a(n) = (4*n^3+6*n^2-13*n+36)/6 for n>0 and even.
a(n) = (4*n^3+18*n^2+11*n+3)/6 for n>0 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>7.
G.f.: (1+5*x+2*x^2+25*x^3-7*x^4+7*x^5+4*x^6-5*x^7) / ((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=275; 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. A271091.
Sequence in context: A292120 A099437 A368373 * A271291 A271085 A271279
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 30 2016
STATUS
approved