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.

A283651 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 705", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 7, 11, 7, 47, 31, 191, 127, 767, 511, 3071, 2047, 12287, 8191, 49151, 32767, 196607, 131071, 786431, 524287, 3145727, 2097151, 12582911, 8388607, 50331647, 33554431, 201326591, 134217727, 805306367, 536870911, 3221225471, 2147483647, 12884901887
Offset: 0

Views

Author

Robert Price, Mar 12 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. A283648, A283649, A283650.

Programs

  • 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]], 2], {i ,1, stages - 1}]

Formula

Conjectures from Colin Barker, Mar 14 2017: (Start)
G.f.: (1 - x + 3*x^2 + 8*x^3 - 32*x^4 + 24*x^5) / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
a(n) = (-2 - (-2)^n + 2^(1+n)) / 2 for n>2.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>5.
(End)

A283649 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 705", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 111, 1011, 111, 101111, 11111, 10111111, 1111111, 1011111111, 111111111, 101111111111, 11111111111, 10111111111111, 1111111111111, 1011111111111111, 111111111111111, 101111111111111111, 11111111111111111, 10111111111111111111, 1111111111111111111
Offset: 0

Views

Author

Robert Price, Mar 12 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. A283648, A283650, A283651.

Programs

  • 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}]

Formula

Conjectures from Colin Barker, Mar 14 2017: (Start)
G.f.: (1 - x + 11*x^2 + 1000*x^3 - 12000*x^4 + 11000*x^5) / ((1 - x)*(1 - 10*x)*(1 + 10*x)).
a(n) = (-10 + 23*2^(1+n)*5^n - 9*(-2)^n*5^(1+n)) / 90 for n>2.
a(n) = a(n-1) + 100*a(n-2) - 100*a(n-3) for n>5.
(End)

A283650 Decimal 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 705", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 7, 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

Views

Author

Robert Price, Mar 12 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. A283648, A283649, A283651.

Programs

  • 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[1, i]], 2], {i, 1, stages - 1}]

Formula

Conjectures from Colin Barker, Mar 14 2017: (Start)
G.f.: (1 - 2*x + 6*x^2 + x^3 - 5*x^4 + 6*x^5) / ((1 - x)*(1 + x)*(1 - 2*x)).
a(n) = 2*(2^n - 2) for n>2 and even.
a(n) = 2^(n+1) - 3 for n>2 and odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>5.
(End)
Showing 1-3 of 3 results.

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