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.

A269713 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 20", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 13, 25, 45, 69, 113, 141, 193, 249, 353, 409, 513, 625, 837, 897, 1013, 1133, 1365, 1485, 1717, 1957, 2421, 2541, 2773, 3013, 3477, 3717, 4181, 4661, 5593, 5717, 5961, 6209, 6697, 6945, 7433, 7929, 8905, 9153, 9641, 10137, 11113, 11609, 12585, 13577
Offset: 0

Views

Author

Robert Price, Mar 04 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Rules 28, 52, 60, 148, 156, 180, 188, 532, 540, 564, 572, 660, 668, 692 and 700 also generate this sequence.

References

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

Links

Crossrefs

Cf. A269711.

Programs

  • Mathematica
    rule=20; stages=300;
ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
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 *)

A269714 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 20", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

3, 4, 4, 8, 4, 20, -16, 24, 4, 48, -48, 48, 8, 100, -152, 56, 4, 112, -112, 112, 8, 224, -344, 112, 8, 224, -224, 224, 16, 452, -808, 120, 4, 240, -240, 240, 8, 480, -728, 240, 8, 480, -480, 480, 16, 960, -1704, 240, 8, 480, -480, 480, 16, 960, -1456, 480
Offset: 0

Views

Author

Robert Price, Mar 04 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Rules 28, 52, 60, 148, 156, 180, 188, 532, 540, 564, 572, 660, 668, 692 and 700 also generate this sequence.

References

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

Links

Crossrefs

Cf. A269711.

Programs

  • Mathematica
    rule=20; stages=300;
ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
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 *)

A269712 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 20", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 4, 12, 28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, 65532, 131068
Offset: 0

Views

Author

Robert Price, Mar 04 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Rules 28, 52, 60, 148, 156, 180, 188, 532, 540, 564, 572, 660, 668, 692 and 700 also generate this sequence.
Apparently a duplicate of A173033. - R. J. Mathar, Mar 09 2016

References

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

Links

Crossrefs

Cf. A269711.

Programs

  • Mathematica
    rule=20; stages=300;
ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
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 08 2016: (Start)
a(n) = 4*(2^n-1) =A028399(n+2) for n>0.
a(n) = 3*a(n-1)-2*a(n-2) for n>2.
G.f.: (1+x+2*x^2) / ((1-x)*(1-2*x)).
(End)

Extensions

a(9)-a(15) from Lars Blomberg, Apr 15 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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