Search: a280373 -id:a280373
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A280371
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Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 260", based on the 5-celled von Neumann neighborhood.
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+10
4
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1, 10, 101, 1000, 10000, 101000, 1010100, 10000010, 100000001, 1010000000, 10101000000, 100000100000, 1000000000000, 10100000000000, 101010000000000, 1000001000000000, 10000000000000000, 101000000000000000, 1010100000000000000, 10000010000000000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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Initialized with a single black (ON) cell at stage zero.
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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LINKS
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FORMULA
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a(n) = 10000*a(n-4) for n>3.
G.f.: (1 + 10*x + 101*x^2 + 1000*x^3 + 1000*x^5 + 100*x^6 + 10*x^7 + x^8 - 10000*x^12) / ((1 - 10*x)*(1 + 10*x)*(1 + 100*x^2)).
(End)
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MATHEMATICA
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CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 260; 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[1, i]], 10], {i, 1, stages - 1}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A280372
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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 260", based on the 5-celled von Neumann neighborhood.
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+10
4
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1, 1, 101, 1, 1, 101, 10101, 1000001, 100000001, 101, 10101, 1000001, 1, 101, 10101, 1000001, 1, 101, 10101, 1000001, 1, 101, 10101, 1000001, 1, 101, 10101, 1000001, 1, 101, 10101, 1000001, 1, 101, 10101, 1000001, 1, 101, 10101, 1000001, 1, 101, 10101
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Initialized with a single black (ON) cell at stage zero.
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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LINKS
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FORMULA
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Empirical g.f.: (1 + x + 101*x^2 + x^3 + 100*x^5 + 10000*x^6 + 1000000*x^7 + 100000000*x^8 -100000000*x^12) / ((1 -x)*(1 + x)*(1 + x^2)). - Colin Barker, Jan 02 2017
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MATHEMATICA
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CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 260; 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}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A280374
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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 260", based on the 5-celled von Neumann neighborhood.
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+10
4
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1, 1, 5, 1, 1, 5, 21, 65, 257, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5, 21, 65, 1, 5
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Initialized with a single black (ON) cell at stage zero.
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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LINKS
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FORMULA
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Empirical g.f.: (1 + x + 5*x^2 + x^3 + 4*x^5 + 16*x^6 + 64*x^7 + 256*x^8 - 256*x^12) / ((1 - x)*(1 + x)*(1 + x^2)). - Colin Barker, Jan 02 2017
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MATHEMATICA
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CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 260; 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}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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