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.

A118173 Decimal representation of n-th iteration of the Rule 188 elementary cellular automaton starting with a single black cell.

Original entry on oeis.org

1, 3, 5, 15, 29, 55, 93, 247, 477, 887, 1501, 3959, 7645, 14199, 24029, 63351, 122333, 227191, 384477, 1013623, 1957341, 3635063, 6151645, 16217975, 31317469, 58161015, 98426333, 259487607, 501079517, 930576247, 1574821341, 4151801719, 8017272285
Offset: 0

Views

Author

Eric W. Weisstein, Apr 13 2006

Keywords

Examples

			1; --> 1
0, 1, 1; --> 3
0, 0, 1, 0, 1; --> 5
0, 0, 0, 1, 1, 1, 1; --> 15
0, 0, 0, 0, 1, 1, 1, 0, 1; --> 29

Links

Crossrefs

Cf. A118174, A265427.

Programs

  • Mathematica
    clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[188, {{1}, 0}, 32]] (* Michael De Vlieger, Oct 08 2015 *)
RecurrenceTable[{a[n+6]==a[n+4] + 16*a[n+2] - 16*a[n], a[0]==1, a[1]==3, a[2]==5, a[3]==15, a[4]==29, a[5]==55}, a, {n,0,100}] (* _G. C. Greubel, Oct 08 2015 *)
  • PARI
    Vec(-(8*x^5-8*x^4-12*x^3-4*x^2-3*x-1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x^2+1)) + O(x^40)) \\ Colin Barker, Oct 08 2015
  • Python
    print([28*4**n//15 + 2**n - (28*2**n//15)*2**n for n in range(50)]) # Karl V. Keller, Jr., Nov 11 2021

Formula

a(n) = (1/30)*(-14 + 3*i*(2*i)^n + 55*2^n) for n odd,
a(n) = (1/15)*(-13 + 3*(2*i)^n + 25*2^n) for n even, where i = sqrt(-1).
From Colin Barker, Oct 08 2015: (Start)
G.f.: -(8*x^5-8*x^4-12*x^3-4*x^2-3*x-1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x^2+1)).
a(n) = a(n-2) + 16*a(n-4) - 16*a(n-6) for n>5. (End)
E.g.f.: (1/15)*(6*sinh(x) + (5/2)*sinh(2x) + 25*exp(2x) - 13*exp(x)) + (1/10)*(2*cos(2x)-sin(2x)). - G. C. Greubel, Oct 08 2015
a(n) = floor(28*4^n/15) + 2^n - floor(28*2^n/15)*2^n. - Karl V. Keller, Jr., Nov 11 2021

A265428 Number of ON (black) cells in the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 2, 2, 4, 4, 5, 5, 7, 7, 8, 8, 10, 10, 11, 11, 13, 13, 14, 14, 16, 16, 17, 17, 19, 19, 20, 20, 22, 22, 23, 23, 25, 25, 26, 26, 28, 28, 29, 29, 31, 31, 32, 32, 34, 34, 35, 35, 37, 37, 38, 38, 40, 40, 41, 41, 43, 43, 44, 44, 46, 46, 47, 47, 49, 49, 50, 50
Offset: 0

Views

Author

Robert Price, Dec 08 2015

Keywords

Examples

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 12 rows, replacing "0" with ".", ignoring "0" outside of range of 1's for better visibility of ON cells, with total number of 1's in the row to the left of the chart:
1  =   1
2  =   1 1
2  =   1 . 1
4  =   1 1 1 1
4  =   1 1 1 . 1
5  =   1 1 . 1 1 1
5  =   1 . 1 1 1 . 1
7  =   1 1 1 1 . 1 1 1
7  =   1 1 1 . 1 1 1 . 1
8  =   1 1 . 1 1 1 . 1 1 1
8  =   1 . 1 1 1 . 1 1 1 . 1
10 =   1 1 1 1 . 1 1 1 . 1 1 1
10 =   1 1 1 . 1 1 1 . 1 1 1 . 1
(End)

References

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

Links

Crossrefs

Cf. A118174.

Programs

  • Mathematica
    rule = 188; 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}]
Count[#, n_ /; n == 1] & /@ CellularAutomaton[188, {{1}, 0}, 66] (* Michael De Vlieger, Dec 09 2015 *)

Formula

Conjectures from Colin Barker, Dec 09 2015 and Apr 16 2019: (Start)
a(n) = (1/8)*(6*n-3*(-1)^n+(1-i)*(-i)^n+(1+i)*i^n+9) where i = sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 4.
G.f.: (1+x+2*x^3-x^4) / ((1-x)^2*(1+x)*(1+x^2)). (End)
a(n) = (6*n+9+2*cos(n*Pi/2)-3*cos(n*Pi)-2*sin(n*Pi/2))/8. - Wesley Ivan Hurt, Oct 02 2017

A265429 Total number of ON (black) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 5, 9, 13, 18, 23, 30, 37, 45, 53, 63, 73, 84, 95, 108, 121, 135, 149, 165, 181, 198, 215, 234, 253, 273, 293, 315, 337, 360, 383, 408, 433, 459, 485, 513, 541, 570, 599, 630, 661, 693, 725, 759, 793, 828, 863, 900, 937, 975, 1013, 1053, 1093, 1134
Offset: 0

Views

Author

Robert Price, Dec 08 2015

Keywords

Examples

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 12 rows, replacing "0" with ".", ignoring "0" outside of range of 1's for better visibility of ON cells, followed by total number of ON cells per row, and running total up to that row:
1                          =  1 ->   1
1 1                        =  2 ->   3
1 . 1                      =  2 ->   5
1 1 1 1                    =  4 ->   9
1 1 1 . 1                  =  4 ->  13
1 1 . 1 1 1                =  5 ->  18
1 . 1 1 1 . 1              =  5 ->  23
1 1 1 1 . 1 1 1            =  7 ->  30
1 1 1 . 1 1 1 . 1          =  7 ->  37
1 1 . 1 1 1 . 1 1 1        =  8 ->  45
1 . 1 1 1 . 1 1 1 . 1      =  8 ->  53
1 1 1 1 . 1 1 1 . 1 1 1    = 10 ->  63
1 1 1 . 1 1 1 . 1 1 1 . 1  = 10 ->  72
(End)

References

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

Links

Crossrefs

Cf. A118174.

Programs

  • Mathematica
    rule = 188; 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}]
Accumulate[Count[#, n_ /; n == 1] & /@ CellularAutomaton[188, {{1}, 0}, 53]] (* Michael De Vlieger, Dec 09 2015 *)

Formula

Conjectures from Colin Barker, Dec 09 2015 and Apr 16 2019: (Start)
a(n) = (1/16)*(6*n^2 + 24*n - 3*(-1)^n + 2*(-i)^n + 2*i^n + 15) where i = sqrt(-1).
G.f.: (1 + x + 2*x^3 - x^4) / ((1-x)^3*(1+x)*(1+x^2)).
(End)
Conjecture: the sequence consists of all numbers k > 0 such that floor(sqrt(8*(k+1)/3)) != floor(sqrt(8*k/3)). - Gevorg Hmayakyan, Sep 01 2019

A265427 Binary representation of the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 11, 101, 1111, 11101, 110111, 1011101, 11110111, 111011101, 1101110111, 10111011101, 111101110111, 1110111011101, 11011101110111, 101110111011101, 1111011101110111, 11101110111011101, 110111011101110111, 1011101110111011101, 11110111011101110111
Offset: 0

Views

Author

Robert Price, Dec 08 2015

Keywords

Examples

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 12 rows:
1
1 1
1 0 1
1 1 1 1
1 1 1 0 1
1 1 0 1 1 1
1 0 1 1 1 0 1
1 1 1 1 0 1 1 1
1 1 1 0 1 1 1 0 1
1 1 0 1 1 1 0 1 1 1
1 0 1 1 1 0 1 1 1 0 1
1 1 1 1 0 1 1 1 0 1 1 1
1 1 1 0 1 1 1 0 1 1 1 0 1
(End)

References

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

Links

Crossrefs

Cf. A118173, A118174.

Programs

  • Mathematica
    rule = 188; 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}]

Formula

Conjectures from Colin Barker, Dec 09 2015 and Apr 18 2019: (Start)
a(n) = a(n-2) + 10000*a(n-4) - 10000*a(n-6) for n>5.
G.f.: (1+11*x+100*x^2+1100*x^3+1000*x^4-1000*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)*(1+100*x^2)).
(End)
Conjecture: a(n) = floor(11100*100^n/9999) + 10^n - floor(11100*10^n/9999)*10^n. - Karl V. Keller, Jr., Dec 15 2021

A265430 Number of OFF (white) cells in the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 1, 3, 3, 5, 6, 8, 8, 10, 11, 13, 13, 15, 16, 18, 18, 20, 21, 23, 23, 25, 26, 28, 28, 30, 31, 33, 33, 35, 36, 38, 38, 40, 41, 43, 43, 45, 46, 48, 48, 50, 51, 53, 53, 55, 56, 58, 58, 60, 61, 63, 63, 65, 66, 68, 68, 70, 71, 73, 73, 75, 76, 78, 78, 80, 81, 83
Offset: 0

Views

Author

Robert Price, Dec 08 2015

Keywords

Examples

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 12 rows, replacing "1" with "." for better visibility of OFF cells, with total number of 0's in the row to the right of the chart:
                      .                        =  0
                    . . 0                      =  1
                  . 0 . 0 0                    =  3
                . . . . 0 0 0                  =  3
              . . . 0 . 0 0 0 0                =  5
            . . 0 . . . 0 0 0 0 0              =  6
          . 0 . . . 0 . 0 0 0 0 0 0            =  8
        . . . . 0 . . . 0 0 0 0 0 0 0          =  8
      . . . 0 . . . 0 . 0 0 0 0 0 0 0 0        = 10
    . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0      = 11
  . 0 . . . 0 . . . 0 . 0 0 0 0 0 0 0 0 0 0    = 13
. . . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 0 0  = 13
(End)

References

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

Links

Crossrefs

Cf. A118174.

Programs

  • Mathematica
    lim = 134; a = {}; Do[AppendTo[a, Take[#[[k]], 2 (k - 1) + 1]], {k, Floor[Length[#]/2]}] &@ CellularAutomaton[188, {{1}, 0}, lim]; Count[#, n_ /; n == 0] & /@ a (* Michael De Vlieger, Dec 09 2015 *)

Formula

Conjectures from Colin Barker, Dec 09 2015 and Apr 16 2019: (Start)
a(n) = 1/8*(10*n+3*(-1)^n-(1-i)*(-i)^n-(1+i)*i^n-1) where i = sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>4.
G.f.: x*(1+2*x+2*x^3) / ((1-x)^2*(1+x)*(1+x^2)).
(End)

A265431 Total number of OFF (white) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 1, 4, 7, 12, 18, 26, 34, 44, 55, 68, 81, 96, 112, 130, 148, 168, 189, 212, 235, 260, 286, 314, 342, 372, 403, 436, 469, 504, 540, 578, 616, 656, 697, 740, 783, 828, 874, 922, 970, 1020, 1071, 1124, 1177, 1232, 1288, 1346, 1404, 1464, 1525, 1588, 1651
Offset: 0

Views

Author

Robert Price, Dec 08 2015

Keywords

Examples

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 12 rows, replacing "1" 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                      =  1 ->  1
                  . 0 . 0 0                    =  3 ->  4
                . . . . 0 0 0                  =  3 ->  7
              . . . 0 . 0 0 0 0                =  5 -> 12
            . . 0 . . . 0 0 0 0 0              =  6 -> 18
          . 0 . . . 0 . 0 0 0 0 0 0            =  8 -> 26
        . . . . 0 . . . 0 0 0 0 0 0 0          =  8 -> 34
      . . . 0 . . . 0 . 0 0 0 0 0 0 0 0        = 10 -> 44
    . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0      = 11 -> 55
  . 0 . . . 0 . . . 0 . 0 0 0 0 0 0 0 0 0 0    = 13 -> 68
. . . . 0 . . . 0 . . . 0 0 0 0 0 0 0 0 0 0 0  = 13 -> 81
(End)

References

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

Links

Crossrefs

Cf. A118174.

Programs

  • Mathematica
    lim = 104; a = {}; Do[AppendTo[a, Take[#[[k]], 2 (k - 1) + 1]], {k, Floor[Length[#]/2]}] &@ CellularAutomaton[188, {{1}, 0}, lim]; Accumulate[Count[#, n_ /; n == 0] & /@ a] (* Michael De Vlieger, Dec 09 2015 *)

Formula

Conjectures from Colin Barker, Dec 09 2015 and Apr 16 2019: (Start)
a(n) = 1/16*(10*n^2+8*n+3*(-1)^n-2*(-i)^n-2*i^n+1) where i = sqrt(-1).
G.f.: x*(1+2*x+2*x^3) / ((1-x)^3*(1+x)*(1+x^2)).
(End)
Showing 1-6 of 6 results.

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