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A282985
Binary 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 541", based on the 5-celled von Neumann neighborhood.
4
1, 1, 1, 1001, 1001, 1001, 1101001, 10001001, 1101001, 1100001001, 10111101001, 11110001001, 10001101001, 10011100001001, 10010111101001, 10011110001001, 11001010001101001, 100101011100001001, 10000010111101001, 11011110011110001001, 100010001010001101001
OFFSET
0,4
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 Chai Wah Wu, May 06 2024: (Start)
a(n) = 10001*a(n-4) - 10000*a(n-8) for n > 33.
G.f.: (-1000000000000000000000*x^33 - 910000000000000000000*x^32 + 899900091000000000000000000*x^31 + 8909991000000000000000000*x^30 - 10901099000000000000000000*x^29 + 990110000000000000000000*x^28 - 899889891000000000000000000*x^27 - 8920890900000000000000000*x^26 + 10901980100000000000000000*x^25 - 989089010000000000000000*x^24 - 10121010000000000000000*x^23 + 10901100000000000000000*x^22 - 990999000000000000000*x^21 - 10011000000000000000*x^20 + 10911100000000000000*x^19 - 90010000000000000*x^18 - 9000000000000*x^17 + 10901000000000000*x^16 - 101000000000000*x^15 - 91100000000000*x^14 - 990000000000*x^13 - 1000000000*x^12 - 88900000000*x^11 - 900000000*x^10 + 1090000000*x^9 - 8900000*x^8 - 10000*x^7 + 1091000*x^6 - 9000*x^5 - 9000*x^4 + 1001*x^3 + x^2 + x + 1)/(10000*x^8 - 10001*x^4 + 1). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 541; 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, Feb 26 2017
STATUS
approved