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.

A277954 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 14", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 6, 14, 26, 58, 106, 234, 426, 938, 1706, 3754, 6826, 15018, 27306, 60074, 109226, 240298, 436906, 961194, 1747626, 3844778, 6990506, 15379114, 27962026, 61516458, 111848106, 246065834, 447392426, 984263338, 1789569706, 3937053354, 7158278826
Offset: 0

Views

Author

Robert Price, Nov 05 2016

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. A277952, A277953, A277955.

Programs

  • 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[1,i]],2], {i,1,stages-1}]
LinearRecurrence[{1, 4, -4}, {1, 3, 6}, 31] (* or *)
CoefficientList[ Series[(1 + 2x - x^2)/(1 - x - 4x^2 + 4x^3), {x, 0, 31}], x] (* Robert G. Wilson v, Nov 05 2016 *)

Formula

Conjectures from Colin Barker, Nov 06 2016: (Start)
G.f.: (1+2*x-x^2) / ((1-x)*(1-2*x)*(1+2*x)).
a(n) = a(n-1)+4*a(n-2)-4*a(n-3) for n>2.
a(n) = (-8-(-2)^n+21*2^n)/12. (End)

A277953 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 14", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 11, 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, Nov 05 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Essentially the same as A267051. - R. J. Mathar, Nov 09 2016

References

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

Links

Crossrefs

Cf. A277952, A277954, A277955.

Programs

  • 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]],10], {i,1,stages-1}]

Formula

Conjectures from Colin Barker, Nov 06 2016: (Start)
G.f.: (1+x-100*x^2) / ((1-x)*(1+x)*(1-10*x)).
a(n) = 10*a(n-1)+a(n-2)-10*a(n-3) for n>2.
a(n) = (539-450*(-1)^n+10^(1+n))/99. (End)

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)
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.