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A290194
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 705", based on the 5-celled von Neumann neighborhood.
4
1, 2, 5, 13, 28, 61, 124, 253, 508, 1021, 2044, 4093, 8188, 16381, 32764, 65533, 131068, 262141, 524284, 1048573, 2097148, 4194301, 8388604, 16777213, 33554428, 67108861, 134217724, 268435453, 536870908, 1073741821, 2147483644, 4294967293, 8589934588
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
Conjecture: a(n) = Fibonacci(2*n+1) if n <= 3, for n > 3, a(n) = 2*a(n-1) + 2 if n is even, a(n) = 2*a(n-1) + 5 if n is odd. It would follow that a(n) = 2^(n+1) - 4 + (n mod 2) for n >= 3. - David A. Corneth, Jul 23 2017
From Chai Wah Wu, Nov 01 2018: (Start)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n > 5 (conjectured).
G.f.: (2*x^5 + x^4 + 3*x^3 + 1)/((x - 1)*(x + 1)*(2*x - 1)) (conjectured). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 705; 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, Jul 23 2017
STATUS
approved