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 1-2 of 2 results.

A277955 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 14", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 3, 7, 11, 23, 43, 87, 171, 343, 683, 1367, 2731, 5463, 10923, 21847, 43691, 87383, 174763, 349527, 699051, 1398103, 2796203, 5592407, 11184811, 22369623, 44739243, 89478487, 178956971, 357913943, 715827883, 1431655767, 2863311531, 5726623063
Offset: 0

Views

Author

Robert Price, Nov 05 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Essentially the same as A267052. - R. J. Mathar, Nov 09 2016

References

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

Links

Crossrefs

Cf. A277952, A277953, A277954.
Cf. A010673, A001045, A007583, A163834.
Cf. A078008.

Programs

  • Magma
    I:=[1,3,3]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-2)-2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Nov 06 2016
  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=14; 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}]
LinearRecurrence[{2, 1, -2}, {1, 3, 3}, 32] (* or *)
CoefficientList[ Series[(1 + x - 4x^2)/(1 - 2x - x^2 + 2x^3), {x, 0, 31}], x] (* Robert G. Wilson v, Nov 05 2016 *)

Formula

G.f.: (1 + x - 4*x^2)/(1 - 2*x - x^2 + 2*x^3). - Robert G. Wilson v, Nov 05 2016
From Colin Barker, Nov 06 2016: (Start)
a(n) = (3 - 2*(-1)^n + 2^(1+n))/3.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. (End)
From Paul Curtz, May 08 2024: (Start)
a(2*n) = A007583(n). a(2*n+1) = A163834(n+1).
a(n) = A001045(n+1) + A010673(n).
a(n) = a(n-1) + 2*A078008(n-1). (End)

A267051 Binary representation of the n-th iteration of the "Rule 92" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 11, 111, 1011, 10111, 101011, 1010111, 10101011, 101010111, 1010101011, 10101010111, 101010101011, 1010101010111, 10101010101011, 101010101010111, 1010101010101011, 10101010101010111, 101010101010101011, 1010101010101010111, 10101010101010101011
Offset: 0

Views

Author

Robert Price, Jan 09 2016

Keywords

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A267050, A267052.

Programs

  • Mathematica
    rule=92; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]]],{k,1,rows}]   (* Binary Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 10 2016 and Apr 17 2019: (Start)
a(n) = (539+450*(-1)^n+10^(2+n))/99 for n>0.
a(n) = 10*a(n-1)+a(n-2)-10*a(n-3) for n>3.
G.f.: (1+x-100*x^3) / ((1-x)*(1+x)*(1-10*x)).
(End)
Showing 1-2 of 2 results.

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