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.

A273336 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 22, 70, 150, 270, 438, 662, 950, 1310, 1750, 2278, 2902, 3630, 4470, 5430, 6518, 7742, 9110, 10630, 12310, 14158, 16182, 18390, 20790, 23390, 26198, 29222, 32470, 35950, 39670, 43638, 47862, 52350, 57110, 62150, 67478, 73102, 79030, 85270, 91830, 98718
Offset: 0

Views

Author

Robert Price, May 20 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. A273334.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=657; 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 *)

Formula

Conjectures from Colin Barker, May 20 2016: (Start)
a(n) = 2/3*(2*n^3+6*n^2+4*n-15) for n>1.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>5.
G.f.: (1+x+8*x^2+8*x^3-17*x^4+7*x^5) / (1-x)^4.
(End)

A273337 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

3, 13, 31, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432
Offset: 0

Views

Author

Robert Price, May 20 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. A273334.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=657; 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 *)

Formula

Conjectures from Colin Barker, May 20 2016: (Start)
a(n) = 8*(1+n) for n>2.
a(n) = 2*a(n-1)-a(n-2) for n>4.
G.f.: (3+7*x+8*x^2-17*x^3+7*x^4) / (1-x)^2.
(End)

A273335 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 4, 48, 224, 960, 3968, 16128, 65024, 261120, 1046528, 4190208, 16769024, 67092480, 268402688, 1073676288, 4294836224
Offset: 0

Views

Author

Robert Price, May 20 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Conjecture: Rule 665 also generates this sequence. - Lars Blomberg, Jul 18 2016
Seems to differ from A271061 only at n=1. - R. J. Mathar, Mar 27 2025

References

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

Links

Crossrefs

Cf. A273334.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=657; 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, May 20 2016: (Start)
a(n) = 2^(n+2)*(2^n-1) for n>1.
a(n) = 6*a(n-1)-8*a(n-2) for n>3.
G.f.: (1-2*x+32*x^2-32*x^3) / ((1-2*x)*(1-4*x)).
(End)

Extensions

a(8)-a(15) from Lars Blomberg, Jul 18 2016
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