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.

A270219 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 10, 38, 47, 131, 152, 292, 317, 577, 614, 994, 1043, 1575, 1660, 2280, 2369, 3237, 3338, 4454, 4567, 5963, 6112, 7724, 7885, 9841, 10038, 12274, 12507, 15055, 15396, 18000, 18345, 21453, 21810, 25422, 25791, 29939, 30344, 34964, 35381, 40601, 41054
Offset: 0

Views

Author

Robert Price, Mar 13 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. A270217.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=129; 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}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)

A270220 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

3, 1, 23, -19, 75, -63, 119, -115, 235, -223, 343, -331, 483, -447, 535, -531, 779, -767, 1015, -1003, 1283, -1247, 1463, -1451, 1795, -1759, 2039, -2003, 2315, -2207, 2263, -2259, 2763, -2751, 3255, -3243, 3779, -3743, 4215, -4203, 4803, -4767, 5303, -5267
Offset: 0

Views

Author

Robert Price, Mar 13 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. A270217.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=129; 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}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Table[on[[i+1]]-on[[i]],{i,1,Length[on]-1}] (* Difference at each stage *)

A270218 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 4, 28, 140, 620, 2604, 10668, 43180, 173740, 697004, 2792108, 11176620, 44722860, 178924204, 715762348, 2863180460
Offset: 0

Views

Author

Robert Price, Mar 13 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
It appears Rules 385, 425, 465 and 553 also generate this sequence. - Lars Blomberg, Apr 30 2016 (It would be nice to have a proof! - N. J. A. Sloane, May 09 2016)

References

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

Links

Crossrefs

Cf. A270217.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=129; 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}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

Formula

Conjectures from Colin Barker, Mar 13 2016: (Start)
a(n) = 4*(1-3*2^n+2^(1+2*n))/3.
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.
G.f.: (1-3*x+14*x^2-8*x^3) / ((1-x)*(1-2*x)*(1-4*x)).
(End)
a(n) = 4*A006095(n+1) (conjectured). - Michal Stajszczak, May 20 2020

Extensions

a(8)-a(15) from Lars Blomberg, Apr 30 2016
Showing 1-3 of 3 results.

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

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