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 1-4 of 4 results.

A265720 Binary representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 0, 100, 1100011, 10000, 11110001111, 1000000, 111111000111111, 100000000, 1111111100011111111, 10000000000, 11111111110001111111111, 1000000000000, 111111111111000111111111111, 100000000000000, 1111111111111100011111111111111, 10000000000000000
Offset: 0

Views

Author

Robert Price, Dec 14 2015

Keywords

Comments

Rule 33 also generates this sequence.

Examples

			From _Michael De Vlieger_, Dec 14 2015: (Start)
First 10 rows, replacing leading zeros with ".", the row converted to its binary equivalent at right:
                  1                    =                   1
                . . 0                  =                   0
              . . 1 0 0                =                 100
            1 1 0 0 0 1 1              =             1100011
          . . . . 1 0 0 0 0            =               10000
        1 1 1 1 0 0 0 1 1 1 1          =         11110001111
      . . . . . . 1 0 0 0 0 0 0        =             1000000
    1 1 1 1 1 1 0 0 0 1 1 1 1 1 1      =     111111000111111
  . . . . . . . . 1 0 0 0 0 0 0 0 0    =           100000000
1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1  = 1111111100011111111
(End)

Links

Crossrefs

Cf. A265718, A265721.

Programs

  • Mathematica
    rule = 1; 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}]
  • Python
    print([(10*100**n - 999*10**(n-1) - 1)//9 if n%2 else 10**n for n in range(50)]) # Karl V. Keller, Jr., Aug 25 2021

Formula

From Colin Barker, Dec 14 2015 and Apr 16 2019: (Start)
a(n) = 10101*a(n-2) - 1010100*a(n-4) + 1000000*a(n-6) for n > 5.
G.f.: (1-10001*x^2+1100011*x^3+10000*x^4-1210000*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)*(1-100*x)*(1+100*x)).
(End)
a(n) = (10*100^n - 999*10^(n-1) - 1)/9 for odd n; a(n) = 10^n for even n. - Karl V. Keller, Jr., Aug 25 2021

A265722 Number of ON (black) cells in the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 0, 1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, 24, 1, 28, 1, 32, 1, 36, 1, 40, 1, 44, 1, 48, 1, 52, 1, 56, 1, 60, 1, 64, 1, 68, 1, 72, 1, 76, 1, 80, 1, 84, 1, 88, 1, 92, 1, 96, 1, 100, 1, 104, 1, 108, 1, 112, 1, 116, 1, 120, 1, 124, 1, 128, 1, 132, 1, 136, 1, 140
Offset: 0

Views

Author

Robert Price, Dec 14 2015

Keywords

Examples

			From _Michael De Vlieger_, Dec 14 2015: (Start)
First 12 rows, replacing zeros with "." for better visibility of ON cells, followed by the total number of 1's per row at right:
                      1                        =   1
                    . . .                      =   0
                  . . 1 . .                    =   1
                1 1 . . . 1 1                  =   4
              . . . . 1 . . . .                =   1
            1 1 1 1 . . . 1 1 1 1              =   8
          . . . . . . 1 . . . . . .            =   1
        1 1 1 1 1 1 . . . 1 1 1 1 1 1          =  12
      . . . . . . . . 1 . . . . . . . .        =   1
    1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1      =  16
  . . . . . . . . . . 1 . . . . . . . . . .    =   1
1 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 1  =  20
(End)

References

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

Links

Crossrefs

Cf. A265718, A265720, A265721.

Programs

  • Mathematica
    rule = 1; rows = 30; 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}]

Formula

Conjectures from Colin Barker, Dec 14 2015 and Apr 16 2019: (Start)
a(n) = 1/2*(-2*(-1)^n*n+2*n+3*(-1)^n-1).
a(n) = 2*a(n-2) - a(n-4) for n>3.
G.f.: (1-x^2+4*x^3) / ((1-x)^2*(1+x)^2).
(End)
a(n) = A019425(n), n>1. - R. J. Mathar, Jan 10 2016

A265723 Number of OFF (white) cells in the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 3, 4, 3, 8, 3, 12, 3, 16, 3, 20, 3, 24, 3, 28, 3, 32, 3, 36, 3, 40, 3, 44, 3, 48, 3, 52, 3, 56, 3, 60, 3, 64, 3, 68, 3, 72, 3, 76, 3, 80, 3, 84, 3, 88, 3, 92, 3, 96, 3, 100, 3, 104, 3, 108, 3, 112, 3, 116, 3, 120, 3, 124, 3, 128, 3, 132, 3, 136, 3, 140, 3
Offset: 0

Views

Author

Robert Price, Dec 14 2015

Keywords

Examples

			From _Michael De Vlieger_, Dec 14 2015: (Start)
First 12 rows, replacing ones with "." for better visibility of OFF cells, followed by the total number of 0's per row at right:
                      .                        =  0
                    0 0 0                      =  3
                  0 0 . 0 0                    =  4
                . . 0 0 0 . .                  =  3
              0 0 0 0 . 0 0 0 0                =  8
            . . . . 0 0 0 . . . .              =  3
          0 0 0 0 0 0 . 0 0 0 0 0 0            = 12
        . . . . . . 0 0 0 . . . . . .          =  3
      0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0        = 16
    . . . . . . . . 0 0 0 . . . . . . . .      =  3
  0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0    = 20
. . . . . . . . . . 0 0 0 . . . . . . . . . .  =  3
(End)

References

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

Links

Crossrefs

Cf. A265718, A265720, A265721.

Programs

  • Mathematica
    rows = 71; Count[#, n_ /; n == 0] & /@ Table[Table[Take[CellularAutomaton[1, {{1}, 0}, rows - 1, {All, All}][[k]], {rows - k + 1, rows + k - 1}], {k, 1, rows}][[k]], {k, 1, rows}] (* Michael De Vlieger, Dec 14 2015 *)

Formula

Conjectures from Colin Barker, Dec 15 2015 and Apr 16 2019: (Start)
a(n) = 1/2*((2*(-1)^n+2)*n-3*((-1)^n-1)).
a(n) = 2*a(n-2) - a(n-4) for n>3.
G.f.: x*(3+4*x-3*x^2) / ((1-x)^2*(1+x)^2).
(End)

A265724 Total number of OFF (white) cells after n iterations of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 3, 7, 10, 18, 21, 33, 36, 52, 55, 75, 78, 102, 105, 133, 136, 168, 171, 207, 210, 250, 253, 297, 300, 348, 351, 403, 406, 462, 465, 525, 528, 592, 595, 663, 666, 738, 741, 817, 820, 900, 903, 987, 990, 1078, 1081, 1173, 1176, 1272, 1275, 1375, 1378, 1482
Offset: 0

Views

Author

Robert Price, Dec 14 2015

Keywords

Examples

			From _Michael De Vlieger_, Dec 14 2015: (Start)
First 12 rows, replacing ones with "." for better visibility of OFF cells, followed by the total number of 0's per row, and the running total up to that row:
                      .                        =  0  ->   0
                    0 0 0                      =  3  ->   3
                  0 0 . 0 0                    =  4  ->   7
                . . 0 0 0 . .                  =  3  ->  10
              0 0 0 0 . 0 0 0 0                =  8  ->  18
            . . . . 0 0 0 . . . .              =  3  ->  21
          0 0 0 0 0 0 . 0 0 0 0 0 0            = 12  ->  33
        . . . . . . 0 0 0 . . . . . .          =  3  ->  36
      0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0        = 16  ->  52
    . . . . . . . . 0 0 0 . . . . . . . .      =  3  ->  55
  0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0    = 20  ->  75
. . . . . . . . . . 0 0 0 . . . . . . . . . .  =  3  ->  78
(End)

References

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

Links

Crossrefs

Cf. A265718, A265720, A265721, A267049.

Programs

  • Mathematica
    rows = 53; Accumulate[Count[#, n_ /; n == 0] & /@ Table[Table[Take[CellularAutomaton[1, {{1}, 0}, rows - 1, {All, All}][[k]], {rows - k + 1, rows + k - 1}], {k, rows}][[k]], {k, 1, rows}]] (* Michael De Vlieger, Dec 14 2015 *)

Formula

Conjectures from Colin Barker, Dec 16 2015 and Apr 16 2019: (Start)
a(n) = 1/2*(n^2+(-1)^n*n+4*n-(-1)^n+1).
a(n) = 1/2*(n^2+5*n) for n even.
a(n) = 1/2*(n^2+3*n+2) for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
G.f.: x*(3+4*x-3*x^2) / ((1-x)^3*(1+x)^2).
(End)
Apparently, a(n) = A267049(n) + 4*floor(n/2) - 1 for n>1. - Hugo Pfoertner, Jun 21 2024
Showing 1-4 of 4 results.

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