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 1-4 of 4 results.

A283648 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 705", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 111, 1101, 11100, 111101, 1111100, 11111101, 111111100, 1111111101, 11111111100, 111111111101, 1111111111100, 11111111111101, 111111111111100, 1111111111111101, 11111111111111100, 111111111111111101, 1111111111111111100, 11111111111111111101
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. A283649, 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[1, i]], 10], {i, 1, stages - 1}]

Formula

Conjectures from Colin Barker, Mar 14 2017: (Start)
G.f.: (1 - 10*x + 110*x^2 + x^3 - 21*x^4 + 110*x^5) / ((1 - x)*(1 + x)*(1 - 10*x)).
a(n) = 10*(10^n - 10)/9 for n>2 and even.
a(n) = (10^(n+1) - 91)/9 for n>2 and odd.
a(n) = 10*a(n-1) + a(n-2) - 10*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)

A303611 a(n) = (-1 - (-2)^(n-2)) mod 2^n.

Original entry on oeis.org

2, 1, 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, 8589934591
Offset: 2

Views

Author

Vincenzo Librandi, May 07 2018

Keywords

Comments

A198693 and A083420 interleaved. From 11 onwards, apparently A283651 and A290195 contain the same terms. - Bruno Berselli, May 07 2018

Links

Crossrefs

All terms belong to A052955 and A180516.
Cf. A083420, A198693, A283651, A290195.

Programs

  • Magma
    [IsOdd(n) select 2^(n-2)-1 else 3*2^(n-2)-1: n in [2..40]];
  • Magma
    I:=[2,1,11]; [n le 3 select I[n] else Self(n-1)+4*Self(n-2)-4*Self(n-3): n in [1..35]];
  • Mathematica
    Table[If[OddQ[n], 2^(n - 2) - 1, 3 2^(n - 2) - 1], {n, 2, 80}]
LinearRecurrence[{1, 4, -4}, {2, 1, 11}, 30]
  • PARI
    a(n) = if (n%2, 2^(n-2) - 1, 3*2^(n-2) - 1); \\ Michel Marcus, May 30 2018

Formula

a(n) = 2^(n-2) - 1 for odd n, otherwise a(n) = 3*2^(n-2) - 1, with n>1.
From Bruno Berselli, May 07 2018: (Start)
O.g.f.: x^2*(2 - x + 2*x^2)/((1 - x)*(1 - 2*x)*(1 + 2*x)).
E.g.f.: (1 + 2*x - 4*exp(x) + exp(-2*x) + 2*exp(2*x))/4.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3).
a(n) = (2 + (-1)^n)*2^(n-2) - 1. (End)
Showing 1-4 of 4 results.

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