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.

A290195 Decimal 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 705", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 5, 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, Jul 23 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. A290192, A290193, A290194.

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

Conjecture: For odd n > 3, a(n) = 2^(n-1) - 1, for even n > 3, a(n) = 3*2^(n-1) - 1. - David A. Corneth, Jul 23 2017
From Chai Wah Wu, Nov 01 2018: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n > 5 (conjectured).
G.f.: (16*x^5 - 20*x^4 + 6*x^3 + 1)/((x - 1)*(2*x - 1)*(2*x + 1)) (conjectured). (End)

A290193 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 705", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 101, 1011, 111, 101111, 11111, 10111111, 1111111, 1011111111, 111111111, 101111111111, 11111111111, 10111111111111, 1111111111111, 1011111111111111, 111111111111111, 101111111111111111, 11111111111111111, 10111111111111111111, 1111111111111111111
Offset: 0

Views

Author

Robert Price, Jul 23 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. A290192, A290194, A290195.

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

Conjecture: For even n > 3, binary representation of 3 * 2^(n - 1) - 1. For odd n > 3, binary representation of 2^(n - 1) - 1. - David A. Corneth, Jul 23 2017
From Chai Wah Wu, Nov 01 2018: (Start)
a(n) = a(n-1) + 100*a(n-2) - 100*a(n-3) for n > 5 (conjectured).
G.f.: (10000*x^5 - 10900*x^4 + 910*x^3 + 1)/((x - 1)*(10*x - 1)*(10*x + 1)) (conjectured). (End)

A290194 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 705", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 2, 5, 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, Jul 23 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. A290192, A290193, A290195.

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

Conjecture: a(n) = Fibonacci(2*n+1) if n <= 3, for n > 3, a(n) = 2*a(n-1) + 2 if n is even, a(n) = 2*a(n-1) + 5 if n is odd. It would follow that a(n) = 2^(n+1) - 4 + (n mod 2) for n >= 3. - David A. Corneth, Jul 23 2017
From Chai Wah Wu, Nov 01 2018: (Start)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n > 5 (conjectured).
G.f.: (2*x^5 + x^4 + 3*x^3 + 1)/((x - 1)*(x + 1)*(2*x - 1)) (conjectured). (End)
Showing 1-3 of 3 results.

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