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A278667
Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood.
4
1, 3, 3, 3, 3, 19, 51, 35, 67, 147, 371, 675, 1347, 2707, 5491, 10915, 21827, 43667, 87411, 174755, 349507, 699027, 1398131, 2796195, 5592387, 11184787, 22369651, 44739235, 89478467, 178956947, 357913971, 715827875, 1431655747, 2863311507, 5726623091
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, Nov 26 2016: (Start)
a(n) = 2*a(n-1) + a(n-4) - 2*a(n-5) for n>7.
G.f.: (1 +x -3*x^2 -3*x^3 -4*x^4 +12*x^5 +16*x^6 -64*x^7 +64*x^10) / ((1 -x)*(1 +x)*(1 -2*x)*(1 +x^2)).
(End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=62; 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}];
Table[FromDigits[Part[ca[[i]][[i]], Range[i, 2*i-1]], 2], {i, 1, stages-1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Nov 25 2016
STATUS
approved