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A288827
Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 515", based on the 5-celled von Neumann neighborhood.
4
1, 3, 5, 13, 27, 59, 119, 247, 495, 1007, 2015, 4063, 8127, 16319, 32639, 65407, 130815, 261887, 523775, 1048063, 2096127, 4193279, 8386559, 16775167, 33550335, 67104767, 134209535, 268427263, 536854527, 1073725439, 2147450879, 4294934527, 8589869055
OFFSET
0,2
COMMENTS
Initialized with a single black (ON) cell at stage zero.
FORMULA
From Stefano Spezia, May 25 2022: (Start)
G.f.: (1 - 4*x^2 + 4*x^3 + 2*x^4 - 4*x^5)/((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4) for n > 5.
a(n) = A000225(n+1) - A016116(n) for n > 1. (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 515; 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]], 10], {i, 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jun 17 2017
STATUS
approved