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 1-5 of 5 results.

A070887 Triangle read by rows giving successive states of one-dimensional cellular automaton generated by "Rule 110".

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, May 19 2002

Keywords

Comments

New state of cell is 1 in every case except when the previous states of the cell and its two neighbors were all the same, or when the left neighbor was 1 and the cell and its right neighbor were both 0.
A cellular automaton using Rule 110 with arbitrary inputs is a universal Turing machine.
Row n has length n.
T(n,k) = A075437(n-1,k-1), k=1..n. - Reinhard Zumkeller, Jun 26 2013

Examples

			1;
1,1;
1,1,1;
1,1,0,1;
1,1,1,1,1; ...
		

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 31ff..

Crossrefs

Cf. A047999.
A071049 gives number of ON cells at n-th generation.

Programs

  • Haskell
    a070887 n k = a070887_tabl !! (n-1) !! (k-1)
    a070887_row n = a070887_tabl !! (n-1)
    a070887_tabl = zipWith take [1..] a075437_tabf
    -- Reinhard Zumkeller, Jun 26 2013
  • Maple
    A070887 := proc(n,k)
        option remember;
        local lef,mid,rig ;
        if k < 1 or k > n then
            0;
        elif n = 1 then
            1;
        else
            lef := procname(n-1,k-2) ;
            mid := procname(n-1,k-1) ;
            rig := procname(n-1,k) ;
            if lef = mid and mid = rig then
                0 ;
            elif lef = 1 and mid =0 and rig =0 then
                0;
            else
                1 ;
            end if;
        end if;
    end proc:
    for n from 1 to 12 do
        for k from 1 to n do
            printf("%d ",A070887(n,k)) ;
        end do:
        printf("\n")
    end do: # R. J. Mathar, Feb 18 2015
  • Mathematica
    rows = 14; ca = CellularAutomaton[110, {{1}, 0}, rows-1]; Flatten[ Table[ca[[k, rows-k+1 ;; -1]], {k, 1, rows}]] (* Jean-François Alcover, May 24 2012 *)

Extensions

More terms from Hans Havermann, May 26 2002

A265320 Binary representation of the n-th iteration of the "Rule 110" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 110, 11100, 1101000, 111110000, 11000100000, 1110011000000, 110101110000000, 11111110100000000, 1100000111000000000, 111000011010000000000, 11010001111100000000000, 1111100110001000000000000, 110001011100110000000000000, 11100111101011100000000000000
Offset: 0

Views

Author

Robert Price, Dec 06 2015

Keywords

References

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

Crossrefs

Cf. A075437.

Programs

  • Mathematica
    rule = 110; rows = 20; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]

A265321 Total number of ON (black) cells after n iterations of the "Rule 110" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 6, 9, 14, 17, 22, 28, 36, 41, 47, 55, 63, 71, 82, 93, 106, 115, 126, 137, 150, 164, 180, 194, 208, 221, 234, 251, 273, 293, 309, 326, 350, 369, 383, 402, 427, 445, 465, 490, 514, 533, 557, 588, 615, 641, 665, 687, 719, 750, 778, 802, 831, 865, 895, 926
Offset: 0

Views

Author

Robert Price, Dec 06 2015

Keywords

References

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

Crossrefs

Cf. A075437.

Programs

  • Mathematica
    rule = 110; rows = 30; Table[Total[Take[Table[Total[Table[Take[CellularAutomaton[rule,{{1},0},rows-1,{All,All}][[k]],{rows-k+1,rows+k-1}],{k,1,rows}][[k]]],{k,1,rows}],k]],{k,1,rows}]

A265322 Number of OFF (white) cells in the n-th iteration of the "Rule 110" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 1, 2, 4, 4, 8, 8, 9, 9, 14, 15, 15, 17, 19, 18, 20, 20, 26, 26, 28, 28, 29, 29, 33, 35, 38, 40, 38, 35, 39, 45, 46, 41, 48, 55, 52, 48, 57, 57, 54, 57, 64, 61, 56, 62, 65, 69, 73, 65, 68, 73, 79, 76, 73, 79, 80, 76, 81, 83, 83, 86, 89, 90, 91, 86, 91, 97
Offset: 0

Views

Author

Robert Price, Dec 06 2015

Keywords

References

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

Crossrefs

Cf. A075437.

Programs

  • Mathematica
    A265322list[nmax_]:=Range[1,2nmax+1,2]-Map[Total,CellularAutomaton[110,{{1},0},nmax]];A265322list[100] (* Paolo Xausa, Oct 04 2023 *)

A265323 Total number of OFF (white) cells after n iterations of the "Rule 110" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 1, 3, 7, 11, 19, 27, 36, 45, 59, 74, 89, 106, 125, 143, 163, 183, 209, 235, 263, 291, 320, 349, 382, 417, 455, 495, 533, 568, 607, 652, 698, 739, 787, 842, 894, 942, 999, 1056, 1110, 1167, 1231, 1292, 1348, 1410, 1475, 1544, 1617, 1682, 1750, 1823, 1902
Offset: 0

Views

Author

Robert Price, Dec 06 2015

Keywords

References

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

Crossrefs

Cf. A075437.

Programs

  • Mathematica
    A265323list[nmax_]:=Accumulate[Range[1,2nmax+1,2]-Map[Total,CellularAutomaton[110,{{1},0},nmax]]];A265323list[100] (* Paolo Xausa, Oct 04 2023 *)
Showing 1-5 of 5 results.