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.

A269717 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 22", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 6, 14, 34, 50, 94, 126, 214, 246, 338, 402, 586, 650, 834, 962, 1330, 1394, 1582, 1710, 2086, 2214, 2590, 2846, 3598, 3726, 4102, 4358, 5110, 5366, 6118, 6630, 8134, 8262, 8642, 8898, 9658, 9914, 10674, 11186, 12706, 12962, 13722, 14234, 15754, 16266
Offset: 0

Views

Author

Robert Price, Mar 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. A269715.

Programs

  • Mathematica
    rule=22; 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 *)

A269718 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 22", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

4, 3, 12, -4, 28, -12, 56, -56, 60, -28, 120, -120, 120, -56, 240, -304, 124, -60, 248, -248, 248, -120, 496, -624, 248, -120, 496, -496, 496, -240, 992, -1376, 252, -124, 504, -504, 504, -248, 1008, -1264, 504, -248, 1008, -1008, 1008, -496, 2016, -2784
Offset: 0

Views

Author

Robert Price, Mar 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. A269715.

Programs

  • Mathematica
    rule=22; 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 *)

A269716 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 22", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 20, 88, 368, 1504, 6080, 24448, 98048, 392704, 1571840, 6289408, 25161728, 100655104, 402636800, 1610579968
Offset: 0

Views

Author

Robert Price, Mar 04 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Appears to coincide with A093357 after the second term. - R. J. Mathar, Mar 09 2016

References

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

Links

Crossrefs

Cf. A269715.

Programs

  • Mathematica
    rule=22; 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) = 2^(n-1)*(3*2^n-2) for n>1.
a(n) = 6*a(n-1)-8*a(n-2) for n>3.
G.f.: (1+2*x)*(1-3*x+4*x^2) / ((1-2*x)*(1-4*x)).
(End)

Extensions

a(9)-a(15) from Lars Blomberg, Apr 15 2016
Showing 1-3 of 3 results.

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