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.

A284351 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 899", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 111, 1101, 11111, 111101, 1111111, 11111101, 111111111, 1111111101, 11111111111, 111111111101, 1111111111111, 11111111111101, 111111111111111, 1111111111111101, 11111111111111111, 111111111111111101, 1111111111111111111, 11111111111111111101
Offset: 0

Views

Author

Robert Price, Mar 25 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. A284352, A284353, A284354.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 899; 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}]

Formula

Conjectures from Colin Barker, Mar 26 2017: (Start)
G.f.: (1 - 9*x + 100*x^2) / ((1 - x)*(1 + x)*(1 - 10*x)).
a(n) = (10^(n+1) - 1)/9 for n even.
a(n) = (10^(n+1) - 91)/9 for n odd.
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n>2.
(End)

A284353 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 899", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 7, 13, 31, 61, 127, 253, 511, 1021, 2047, 4093, 8191, 16381, 32767, 65533, 131071, 262141, 524287, 1048573, 2097151, 4194301, 8388607, 16777213, 33554431, 67108861, 134217727, 268435453, 536870911, 1073741821, 2147483647, 4294967293, 8589934591
Offset: 0

Views

Author

Robert Price, Mar 25 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
If one begins the Generalized Jacobsthal numbers (A083579) with a(0)=1, instead of a(0)=0, the same sequence will be obtained. - Henrik Lipskoch, Jan 28 2021

References

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

Links

Crossrefs

Cf. A000975, A284351, A284352, A284354.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 899; 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 26 2017: (Start)
G.f.: (1 - x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
a(n) = 2^(n+1) - 1 for n even.
a(n) = 2^(n+1) - 3 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. (End)
Conjecture: For n > 0, a(n) = Sum_{k=0..n-1} C(n,k) * (2-(-1)^k). - Wesley Ivan Hurt, Sep 23 2017
Apparently, a(n) = 6*A000975(n-1) + 1 for n >= 1. - Hugo Pfoertner, Jan 28 2021

A284354 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 899", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 2, 7, 11, 31, 47, 127, 191, 511, 767, 2047, 3071, 8191, 12287, 32767, 49151, 131071, 196607, 524287, 786431, 2097151, 3145727, 8388607, 12582911, 33554431, 50331647, 134217727, 201326591, 536870911, 805306367, 2147483647, 3221225471, 8589934591
Offset: 0

Views

Author

Robert Price, Mar 25 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. A284351, A284352, A284353.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 899; 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 26 2017: (Start)
G.f.: (1 + x + x^2) / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
a(n) = (2^(n+3) - 4)/4 for n even.
a(n) = (3*2^(n+1) - 4)/4 for n odd.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>2.
(End)
Showing 1-3 of 3 results.

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