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.

A077896 Expansion of (1-x)^(-1)/(1+x-2*x^2-2*x^3).

Original entry on oeis.org

1, 0, 3, 0, 7, 0, 15, 0, 31, 0, 63, 0, 127, 0, 255, 0, 511, 0, 1023, 0, 2047, 0, 4095, 0, 8191, 0, 16383, 0, 32767, 0, 65535, 0, 131071, 0, 262143, 0, 524287, 0, 1048575, 0, 2097151, 0, 4194303, 0, 8388607, 0, 16777215, 0, 33554431, 0, 67108863, 0, 134217727, 0, 268435455
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

Also, the 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 276", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, May 25 2017

References

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

Links

Crossrefs

Cf. A287468, A287469, A287470.

Programs

  • Magma
    [(1+(-1)^n)*(2^Floor((n+3)/2)-1)/2: n in [0..60]]; // Vincenzo Librandi, May 26 2017
  • Mathematica
    CoefficientList[Series[(1 - x)^(-1) / (1 + x - 2 x^2 - 2 x^3), {x, 0, 60}], x] (* Vincenzo Librandi, May 26 2017 *)

Formula

G.f.: 1/((1 - x)*(1 + x)*(1 - 2*x^2)). - Bruno Berselli, May 26 2017
a(n) = (1 + (-1)^n)*(2^floor((n + 3)/2) - 1)/2. - Vincenzo Librandi, May 27 2017

A287469 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 276", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 110, 0, 11100, 0, 1111000, 0, 111110000, 0, 11111100000, 0, 1111111000000, 0, 111111110000000, 0, 11111111100000000, 0, 1111111111000000000, 0, 111111111110000000000, 0, 11111111111100000000000, 0, 1111111111111000000000000, 0
Offset: 0

Views

Author

Robert Price, May 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. A287468, A077896, A287470.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 276; 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

From Colin Barker, May 29 2017: (Start)
Conjectures:
G.f.: 1 / ((1 - 10*x)*(1 + 10*x)*(1 - 10*x^2)).
a(n) = (10^(n+1) - 10^(n/2)) / 9 for n even.
a(n) = 0 for n odd.
a(n) = 110*a(n-2) - 1000*a(n-4) for n>3.
(End)

A287470 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 276", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 6, 0, 28, 0, 120, 0, 496, 0, 2016, 0, 8128, 0, 32640, 0, 130816, 0, 523776, 0, 2096128, 0, 8386560, 0, 33550336, 0, 134209536, 0, 536854528, 0, 2147450880, 0, 8589869056, 0, 34359607296, 0, 137438691328, 0, 549755289600, 0, 2199022206976, 0
Offset: 0

Views

Author

Robert Price, May 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. A077896, A287468, A287469.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 276; 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

From Colin Barker, May 29 2017: (Start)
Conjectures:
G.f.: 1 / ((1 - 2*x)*(1 + 2*x)*(1 - 2*x^2)).
a(n) = 2^(n+1) - 2^(n/2) for n even.
a(n) = 0 for n odd.
a(n) = 6*a(n-2) - 8*a(n-4) for n>3.
(End)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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