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.

A169699 Total number of ON cells at stage n of two-dimensional 5-neighbor outer totalistic cellular automaton defined by "Rule 510".

Original entry on oeis.org

1, 5, 12, 25, 28, 56, 56, 113, 60, 120, 120, 240, 120, 240, 240, 481, 124, 248, 248, 496, 248, 496, 496, 992, 248, 496, 496, 992, 496, 992, 992, 1985, 252, 504, 504, 1008, 504, 1008, 1008, 2016, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032, 504, 1008, 1008, 2016
Offset: 0

Views

Author

N. J. A. Sloane, Apr 17 2010

Keywords

Comments

We work on the square grid. Each cell has 4 neighbors, N, S, E, W. If none of your 4 neighbors are ON, your state does not change. If all 4 of your neighbors are ON, your state flips. In all other cases you turn ON. We start with one ON cell.
As observed by Packard and Wolfram (see Fig. 2), a slice along the E-W line shows the successive states of the 1-D CA Rule 126 (see A071035, A071051).

Examples

			When arranged into blocks of sizes 1,1,2,4,8,16,...:
1,
5,
12, 25,
28, 56, 56, 113,
60, 120, 120, 240, 120, 240, 240, 481,
124, 248, 248, 496, 248, 496, 496, 992, 248, 496, 496, 992, 496, 992, 992, 1985,
252, 504, 504, 1008, 504, 1008, 1008, 2016, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032,
..., the initial terms in the rows (after the initial rows) have the form 2^m-4 and the final terms are given by A092440. The row beginning with 2^m-4 is divisible by 2^(m-2)-1 (see formula).

References

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

Links

Crossrefs

Cf. A000120, A071035, A071051, A008574, A092440, A169700.
See A253089 for 9-celled neighborhood version.

Programs

  • Maple
    A000120 := proc(n) add(i,i=convert(n,base,2)) end:
ht:=n->floor(log[2](n));
f:=proc(n) local a,t1;
if n=0 then 1 else
a:=(2^(ht(n)+1)-1)*2^(1+A000120(n));
if 2^log[2](n)=n then a:=a+1; fi; a; fi; end;
[seq(f(n),n=0..65)]; # A169699
  • Mathematica
    Map[Function[Apply[Plus,Flatten[ #1]]], CellularAutomaton[{ 510, {2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},100]]
ArrayPlot /@ CellularAutomaton[{510, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 28]

Formula

For n>0, it is easy to show that if 2^k <= n < 2^(k+1) then a(n) =
(2^(k+1)-1)*2^(1+wt(n)), where wt is the binary weight A000120, except that if n is a power of 2 we must add 1 to the result.

Extensions

Entry revised with more precise definition, formula and additional information, N. J. A. Sloane, Aug 24 2014

A071035 Triangle read by rows giving successive states of cellular automaton generated by "Rule 126".

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1
Offset: 0

Views

Author

Hans Havermann, May 26 2002

Keywords

Comments

Row n has length 2n+1.

Examples

			Triangle begins:
                           1,
                        1, 1, 1,
                     1, 1, 0, 1, 1,
                  1, 1, 1, 1, 1, 1, 1,
               1, 1, 0, 0, 0, 0, 0, 1, 1,
            1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1,
         1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1,
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
   1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,
1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1,
... - _N. J. A. Sloane_, Aug 24 2014

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Links

Crossrefs

See A071051 for number of ON states.
The right half is the triangle A127872.
Cf. A036987 (central column).

Programs

  • Mathematica
    A071035list[rowmax_]:=MapIndexed[ArrayPad[#1, #2-rowmax-1]&,CellularAutomaton[126,{{1},0},rowmax]];A071035list[10] (* Generates 11 rows *) (* Paolo Xausa, Jul 18 2023 *)

Extensions

Corrected by Hans Havermann, Jan 07 2012

A267447 Number of OFF (white) cells in the n-th iteration of the "Rule 129" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 3, 4, 7, 4, 8, 8, 15, 4, 8, 8, 16, 8, 16, 16, 31, 4, 8, 8, 16, 8, 16, 16, 32, 8, 16, 16, 32, 16, 32, 32, 63, 4, 8, 8, 16, 8, 16, 16, 32, 8, 16, 16, 32, 16, 32, 32, 64, 8, 16, 16, 32, 16, 32, 32, 64, 16, 32, 32, 64, 32, 64, 64, 127, 4, 8, 8, 16, 8, 16, 16
Offset: 0

Views

Author

Robert Price, Jan 15 2016

Keywords

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A267417.

Programs

  • Mathematica
    rule=129; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}] (* Number of White cells in stage n *)

Formula

Conjecture: a(n) = A071051(n), n>0. - R. J. Mathar, Jan 17 2016
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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