cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A286770 Binary 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 221", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 10, 0, 1110, 1, 111110, 1, 11111110, 1, 1111111110, 1, 111111111110, 1, 11111111111110, 1, 1111111111111110, 1, 111111111111111110, 1, 11111111111111111110, 1, 1111111111111111111110, 1, 111111111111111111111110, 1, 11111111111111111111111110, 1
Offset: 0

Views

Author

Robert Price, May 14 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
a(n) = A280410(n) for n >= 3 ? - Michel Marcus, May 17 2017

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A286771, A286772, A286773, A280410.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 221; 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}]

Formula

Conjectures from Colin Barker, May 14 2017: (Start)
G.f.: (1 + 10*x - 101*x^2 + 100*x^3 + 101*x^4 - 100*x^6) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
a(n) = 1 for n>2 and even.
a(n) = 10*(10^n - 1)/9 for n>2 and odd.
a(n) = 101*a(n-2) - 100*a(n-4) for n>4.
(End)

A286771 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 221", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 0, 111, 10000, 11111, 1000000, 1111111, 100000000, 111111111, 10000000000, 11111111111, 1000000000000, 1111111111111, 100000000000000, 111111111111111, 10000000000000000, 11111111111111111, 1000000000000000000, 1111111111111111111
Offset: 0

Views

Author

Robert Price, May 14 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A286770, A286772, A286773.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 221; 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}]

Formula

Conjectures from Colin Barker, May 14 2017: (Start)
G.f.: (1 + x - 101*x^2 + 10*x^3 + 10100*x^4 - 10000*x^6) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
a(n) = 10^n for n>2 and even.
a(n) = (10^n - 1)/9 for n>2 and odd.
a(n) = 101*a(n-2) - 100*a(n-4) for n>4.
(End)
It appears that a(n) = A280411(n) for n >= 3. - Michel Marcus, May 20 2017

A286772 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 221", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 2, 0, 14, 1, 62, 1, 254, 1, 1022, 1, 4094, 1, 16382, 1, 65534, 1, 262142, 1, 1048574, 1, 4194302, 1, 16777214, 1, 67108862, 1, 268435454, 1, 1073741822, 1, 4294967294, 1, 17179869182, 1, 68719476734, 1, 274877906942, 1, 1099511627774, 1, 4398046511102, 1
Offset: 0

Views

Author

Robert Price, May 14 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A286770, A286771, A286773.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 221; 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}]

Formula

Conjectures from Colin Barker, May 14 2017: (Start)
G.f.: (1 + 2*x - 5*x^2 + 4*x^3 + 5*x^4 - 4*x^6) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 1 for n>2.
a(n) = 2^(n+1) - 2 for n>2.
a(n) = 5*a(n-2) - 4*a(n-4) for n>4.
(End)
It appears that a(n) = A280412(n) for n >= 3. - Michel Marcus, May 20 2017
Showing 1-3 of 3 results.

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