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.

A277928 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 5", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 2, 0, 15, 0, 63, 0, 255, 0, 1023, 0, 4095, 0, 16383, 0, 65535, 0, 262143, 0, 1048575, 0, 4194303, 0, 16777215, 0, 67108863, 0, 268435455, 0, 1073741823, 0, 4294967295, 0, 17179869183, 0, 68719476735, 0, 274877906943, 0, 1099511627775, 0, 4398046511103, 0
Offset: 0

Views

Author

Robert Price, Nov 04 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. A277926, A277927, A277929.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=5; 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, Nov 05 2016: (Start)
a(n) = 0 for n>1 and even; a(n) = 2^n-1-(-2)^n for n>1 and odd.
a(n) = 5*a(n-2)-4*a(n-4) for n>5.
G.f.: (1+2*x-5*x^2+5*x^3+4*x^4-4*x^5) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)).
(End)

A277926 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 5", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 10, 0, 1111, 0, 111111, 0, 11111111, 0, 1111111111, 0, 111111111111, 0, 11111111111111, 0, 1111111111111111, 0, 111111111111111111, 0, 11111111111111111111, 0, 1111111111111111111111, 0, 111111111111111111111111, 0, 11111111111111111111111111, 0
Offset: 0

Views

Author

Robert Price, Nov 04 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. A277927, A277928, A277929.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=5; 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, Nov 04 2016: (Start)
a(n) = 0 for n>1 and even; a(n) = (10^(n+1)-1)/9 for n>1 and odd.
a(n) = 101*a(n-2)-100*a(n-4) for n>5.
G.f.: (1+10*x-101*x^2+101*x^3+100*x^4-100*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)).
(End)

A277927 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 5", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 0, 1111, 0, 111111, 0, 11111111, 0, 1111111111, 0, 111111111111, 0, 11111111111111, 0, 1111111111111111, 0, 111111111111111111, 0, 11111111111111111111, 0, 1111111111111111111111, 0, 111111111111111111111111, 0, 11111111111111111111111111, 0
Offset: 0

Views

Author

Robert Price, Nov 04 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Appears to be essentially the same as A277926 and A277560. - R. J. Mathar, Nov 09 2016

References

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

Links

Crossrefs

Cf. A277926, A277928, A277929.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=5; 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 04 2016: (Start)
a(n) = 0 for n>1 and even.
a(n) = (10^(n+1)-1)/9 for n>1 and odd.
a(n) = 101*a(n-2)-100*a(n-4) for n>5.
G.f.: (1+x-101*x^2+1010*x^3+100*x^4-1000*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)).
(End)

A277936 Decimal representation of the x-axis, from the left edge to the origin, or from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 7", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 0, 15, 0, 63, 0, 255, 0, 1023, 0, 4095, 0, 16383, 0, 65535, 0, 262143, 0, 1048575, 0, 4194303, 0, 16777215, 0, 67108863, 0, 268435455, 0, 1073741823, 0, 4294967295, 0, 17179869183, 0, 68719476735, 0, 274877906943, 0, 1099511627775, 0, 4398046511103, 0
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 A277929 and A277928. - R. J. Mathar, Nov 09 2016

References

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

Links

Crossrefs

Cf. A277560.

Programs

  • Maple
    A277936:=n->(-1/2-(-2)^n+(-1)^n/2+2^n): 1,seq(A277936(n), n=1..80); # Wesley Ivan Hurt, Jan 24 2017
  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=7; 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, Nov 06 2016: (Start)
G.f.: (1+3*x-5*x^2+4*x^4) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)).
a(n) = 5*a(n-2)-4*a(n-4) for n>4.
a(n) = (-1/2-(-2)^n+(-1)^n/2+2^n) for n>0. (End)
Showing 1-4 of 4 results.

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