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 1-6 of 6 results.

A246027 a(n) = n - A071049(n).

Original entry on oeis.org

-1, -1, -1, 0, -1, 2, 1, 1, 0, 4, 4, 3, 4, 5, 3, 4, 3, 8, 7, 8, 7, 7, 6, 9, 10, 12, 13, 10, 6, 9, 14, 14, 8, 14, 20, 16, 11, 19, 18, 14, 16, 22, 18, 12, 17, 19, 22, 25, 16, 18, 22, 27, 23, 19, 24, 24, 19, 23, 24, 23, 25, 27, 27, 27, 21, 25, 30, 29, 31, 30, 30, 27, 28, 31, 29, 27, 33, 30, 42, 42, 34
Offset: 0

Views

Author

N. J. A. Sloane, Aug 14 2014

Keywords

Comments

Note that this is much larger than the analogous sequence for Rule 30 (see A070952, A246024). This is because it appears that A071049 only grows as c*n, where c is about 3/5, whereas A070952 is roughly equal to n.

Links

Crossrefs

Cf. A071049, A246024, A070952.

A151931 First differences of A071049.

Original entry on oeis.org

1, 1, 1, 0, 2, -2, 2, 1, 2, -3, 1, 2, 0, 0, 3, 0, 2, -4, 2, 0, 2, 1, 2, -2, 0, -1, 0, 4, 5, -2, -4, 1, 7, -5, -5, 5, 6, -7, 2, 5, -1, -5, 5, 7, -4, -1, -2, -2, 10, -1, -3, -4, 5, 5, -4, 1, 6, -3, 0, 2, -1, -1, 1, 1, 7, -3, -4, 2, -1, 2, 1, 4, 0, -2, 3, 3, -5, 4, -11, 1, 9, 0, -1, -4, 1, 5, 0, 10, -7, 2, 1, -1, 2, -1, -2, 1, 10, -5, -5, -1, 1
Offset: 0

Views

Author

N. J. A. Sloane, Aug 10 2009; corrected Oct 25 2009

Keywords

Comments

Net increase in number of ON cells at generation n of 1-D CA using Rule 110.

Links

A070952 Number of 1's in n-th generation of 1-D CA using Rule 30, started with a single 1.

Original entry on oeis.org

1, 3, 3, 6, 4, 9, 5, 12, 7, 12, 11, 14, 12, 19, 13, 22, 15, 19, 20, 24, 21, 23, 23, 28, 26, 27, 26, 33, 30, 34, 31, 39, 26, 39, 29, 46, 32, 44, 38, 45, 47, 41, 45, 49, 38, 55, 42, 51, 44, 53, 43, 59, 52, 60, 49, 65, 57, 60, 56, 69, 61, 70, 59, 78, 64, 56, 65, 69, 69
Offset: 0

Views

Author

N. J. A. Sloane, May 19 2002, Aug 10 2009

Keywords

Comments

Number of 1's in n-th row of triangle in A070950.
Row sums in A070950; a(n) = 2*n + 1 - A070951(n). - Reinhard Zumkeller, Jun 07 2013

Examples

			May be arranged into blocks of length 1,1,2,4,8,16,...:
1,
3,
3, 6,
4, 9, 5, 12,
7, 12, 11, 14, 12, 19, 13, 22,
15, 19, 20, 24, 21, 23, 23, 28, 26, 27, 26, 33, 30, 34, 31, 39,
26, 39, 29, 46, 32, 44, 38, 45, 47, 41, 45, 49, 38, 55, 42, 51,
    44, 53, 43, 59, 52, 60, 49, 65, 57, 60, 56, 69, 61, 70, 59, 78,
64, 56, 65, 69, 69, ...

Links

Crossrefs

This sequence, A110240, and A245549 all describe the same sequence of successive states. See also A269160.
Cf. A071049, A070950, A070951, A151929, A051023.
Cf. A110267 (partial sums), A246023, A246024, A246025, A246026, A246597.
A265703 is an essentially identical sequence.

Programs

  • Haskell
    a070952 = sum . a070950_row  -- Reinhard Zumkeller, Jun 07 2013
  • Mathematica
    Map[Function[Apply[Plus,Flatten[ #1]]], CellularAutomaton[30,{{1},0},100]] (* N. J. A. Sloane, Aug 10 2009 *)
SequenceCount[s, {1,0}] + 2 SequenceCount[s, {0,0,1}] (* gives a(n) where s is the sequence for row n-1 *) (* Trevor Cappallo, May 01 2021 *)

Extensions

More terms from Hans Havermann, May 26 2002
Corrected offset and initial term - N. J. A. Sloane, Jun 07 2013

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..

Links

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

A161903 Convert n into a sequence of binary digits, apply one step of the rule 110 cellular automaton, and interpret the results as a binary integer.

Original entry on oeis.org

0, 3, 6, 7, 12, 15, 14, 13, 24, 27, 30, 31, 28, 31, 26, 25, 48, 51, 54, 55, 60, 63, 62, 61, 56, 59, 62, 63, 52, 55, 50, 49, 96, 99, 102, 103, 108, 111, 110, 109, 120, 123, 126, 127, 124, 127, 122, 121, 112, 115, 118, 119, 124, 127, 126, 125, 104, 107, 110, 111, 100, 103, 98, 97, 192, 195, 198, 199, 204, 207, 206, 205, 216, 219, 222, 223, 220, 223, 218, 217, 240, 243, 246, 247, 252, 255, 254, 253, 248, 251, 254, 255, 244, 247, 242, 241, 224, 227, 230, 231, 236
Offset: 0

Views

Author

Ben Branman, Jan 30 2011

Keywords

Comments

a(a(a(...1))) (n times) gives A006978(n)

Examples

			For n=19, the evolution after one step is
0, 1, 0, 0, 1, 1  (n=19)
1, 1, 0, 1, 1, 1  (a(n)=55)
So a(n)=55.

Links

Crossrefs

Cf. A006978, A070887, A071049, A180001, A186083, A204371.
Cf. also A269160, A269174, A269175.

Programs

  • Mathematica
    a[n_] :=
FromDigits[
  Drop[Part[CellularAutomaton[110, {IntegerDigits[n, 2], 0}], 1], -1],
   2];Table[a[n],{n,0,100}]

Formula

a(n) = A057889(A269174(A057889(n))). - Antti Karttunen, Jun 02 2018

A267518 Number of OFF (white) cells in the n-th iteration of the "Rule 137" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 3, 3, 5, 3, 5, 6, 8, 5, 6, 8, 8, 8, 11, 11, 13, 9, 11, 11, 13, 14, 16, 14, 14, 13, 13, 17, 22, 20, 16, 17, 24, 19, 14, 19, 25, 18, 20, 25, 24, 19, 24, 31, 27, 26, 24, 22, 32, 31, 28, 24, 29, 34, 30, 31, 37, 34, 34, 36, 35, 34, 35, 36, 43, 40, 36, 38, 37
Offset: 0

Views

Author

Robert Price, Jan 16 2016

Keywords

References

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

Links

Crossrefs

Cf. A071049, A267463.

Programs

  • Mathematica
    rule=137; 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 *)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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