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A277955
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 14", based on the 5-celled von Neumann neighborhood.
4
1, 3, 3, 7, 11, 23, 43, 87, 171, 343, 683, 1367, 2731, 5463, 10923, 21847, 43691, 87383, 174763, 349527, 699051, 1398103, 2796203, 5592407, 11184811, 22369623, 44739243, 89478487, 178956971, 357913943, 715827883, 1431655767, 2863311531, 5726623063
OFFSET
0,2
COMMENTS
Initialized with a single black (ON) cell at stage zero.
Essentially the same as A267052. - R. J. Mathar, Nov 09 2016
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
FORMULA
G.f.: (1 + x - 4*x^2)/(1 - 2*x - x^2 + 2*x^3). - Robert G. Wilson v, Nov 05 2016
From Colin Barker, Nov 06 2016: (Start)
a(n) = (3 - 2*(-1)^n + 2^(1+n))/3.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. (End)
From Paul Curtz, May 08 2024: (Start)
a(2*n) = A007583(n). a(2*n+1) = A163834(n+1).
a(n) = A001045(n+1) + A010673(n).
a(n) = a(n-1) + 2*A078008(n-1). (End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=14; 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}]
LinearRecurrence[{2, 1, -2}, {1, 3, 3}, 32] (* or *)
CoefficientList[ Series[(1 + x - 4x^2)/(1 - 2x - x^2 + 2x^3), {x, 0, 31}], x] (* Robert G. Wilson v, Nov 05 2016 *)
PROG
(Magma) I:=[1, 3, 3]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-2)-2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Nov 06 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Nov 05 2016
STATUS
approved