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

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

1,1

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

References

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

Links

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Eric Weisstein's World of Mathematics, Rule 110

Wikipedia, Rule 110

Index entries for sequences related to cellular automata

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A070950, A070886.

Cf. A047999.

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

Sequence in context: A166280 A340371 A340374 * A209635 A355689 A353627

Adjacent sequences: A070884 A070885 A070886 * A070888 A070889 A070890

Keyword

nonn,tabl,nice,easy

Author

N. J. A. Sloane, May 19 2002

Extensions

More terms from Hans Havermann, May 26 2002

Status

approved