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

Author

N. J. A. Sloane, May 19 2002

Keywords

• nonn
• tabl
• nice
• easy

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..

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

Crossrefs

Cf. A070950, A070886.

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