A118109 -id:A118109 - cp's OEIS frontend

A071030 Triangle read by rows giving successive states of cellular automaton generated by "Rule 54".

1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1

Offset: 0

Hans Havermann, May 26 2002

Row n has length 2n+1.

Even rows r sum to r/2 + 1, odd rows r sum to 3r to produce the sequence {1, 3, 2, 6, 3, 9, 4, 12, ...} = A064455(n + 1). - Michael De Vlieger, Oct 07 2015

			From _Michael De Vlieger_, Oct 07 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells:
                        1
                      1 1 1
                    1 . . . 1
                  1 1 1 . 1 1 1
                1 . . . 1 . . . 1
              1 1 1 . 1 1 1 . 1 1 1
            1 . . . 1 . . . 1 . . . 1
          1 1 1 . 1 1 1 . 1 1 1 . 1 1 1
        1 . . . 1 . . . 1 . . . 1 . . . 1
      1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1
    1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1
  1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1
1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Visualization of rows 0 - 31 of Rule 54
S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601--644.
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

Cf. A064455. See A118108 and A118109 for two other versions of this sequence.

clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; clip /@ CellularAutomaton[54, {{1}, 0}, 8] // Flatten (* Michael De Vlieger, Oct 07 2015 *)

Corrected by Hans Havermann, Jan 07 2012

A118108 Decimal representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.

Original entry on oeis.org

1, 7, 17, 119, 273, 1911, 4369, 30583, 69905, 489335, 1118481, 7829367, 17895697, 125269879, 286331153, 2004318071, 4581298449, 32069089143, 73300775185, 513105426295, 1172812402961, 8209686820727, 18764998447377, 131354989131639, 300239975158033

Offset: 0

Eric W. Weisstein, Apr 13 2006

a(1660) is 1000 digits long. - Michael De Vlieger, Oct 07 2015

			From _Michael De Vlieger_, Oct 07 2015: (Start)
First 8 rows, representing ON cells as "1", OFF cells within the bounds of ON cells as "0", interpreted as a binary number at left, the decimal equivalent appearing at right:
                    1 =     1
                  111 =     7
               1 0001 =    17
             111 0111 =   119
          1 0001 0001 =   273
        111 0111 0111 =  1911
     1 0001 0001 0001 =  4369
   111 0111 0111 0111 = 30583
1 0001 0001 0001 0001 = 69905
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..1660
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig 8.
Eric Weisstein's World of Mathematics, Rule 54
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (0,17,0,-16).

See A071030, A118109 for two other versions of this sequence.

clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[54, {{1}, 0}, 27]] (* or *)
Table[If[EvenQ@ n, (4^(n + 2) - 1), 7 (4^(n + 1) - 1)]/15, {n, 0, 27}] (* Michael De Vlieger, Oct 07 2015 *)

print([(16+12*(n%2))*4**n//15 for n in range(30)]) # Karl V. Keller, Jr., Aug 04 2021

a(n) = 7*(4^(n+1)-1)/15 for n odd; a(n) = (4^(n+2)-1)/15 for n even.
From Colin Barker, Oct 08 2015 and Apr 16 2019: (Start)
a(n) = 17*a(n-2) - 16*a(n-4) for n>3.
G.f.: (7*x+1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).
(End)
a(n) = floor((16+12*(n mod 2))*4^n/15). - Karl V. Keller, Jr., Aug 04 2021

a(23)-a(24) from Michael De Vlieger, Oct 07 2015

A265225 Total number of ON (black) cells after n iterations of the "Rule 54" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 6, 12, 15, 24, 28, 40, 45, 60, 66, 84, 91, 112, 120, 144, 153, 180, 190, 220, 231, 264, 276, 312, 325, 364, 378, 420, 435, 480, 496, 544, 561, 612, 630, 684, 703, 760, 780, 840, 861, 924, 946, 1012, 1035, 1104, 1128, 1200, 1225, 1300, 1326, 1404, 1431

Offset: 0

Robert Price, Dec 05 2015

Take the first 2n positive integers and choose n of them such that their sum: a) is divisible by n, and b) is minimal. It seems their sum equals a(n). - Ivan N. Ianakiev, Feb 16 2019

			From _Michael De Vlieger_, Dec 14 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row, and the running total up to that row:
                        1                          =  1 ->  1
                      1 1 1                        =  3 ->  4
                    1 . . . 1                      =  2 ->  6
                  1 1 1 . 1 1 1                    =  6 -> 12
                1 . . . 1 . . . 1                  =  3 -> 15
              1 1 1 . 1 1 1 . 1 1 1                =  9 -> 24
            1 . . . 1 . . . 1 . . . 1              =  4 -> 28
          1 1 1 . 1 1 1 . 1 1 1 . 1 1 1            = 12 -> 40
        1 . . . 1 . . . 1 . . . 1 . . . 1          =  5 -> 45
      1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1        = 15 -> 60
    1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1      =  6 -> 66
  1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1    = 18 -> 84
1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1  =  7 -> 91
(End)

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

Robert Price, Table of n, a(n) for n = 0..999
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche (2021) Vol. 76, No. 1, see p. 301.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A071030, A118108, A118109, A133872.

A265225:=n->1/4*(n+1)*(2*n-(-1)^n+5): seq(A265225(n), n=0..60); # Wesley Ivan Hurt, Dec 25 2016

rule = 54; 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[Total /@ CellularAutomaton[54, {{1}, 0}, 52]]

Conjectures from Colin Barker, Dec 08 2015 and Apr 20 2019: (Start)
a(n) = (n+1)*(2*n -(-1)^n +5)/4.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
G.f.: (1+3*x) / ((1-x)^3*(1+x)^2).
(End)
a(n) = n + 1 + (n+1) * floor((n+1)/2), conjectured. - Wesley Ivan Hurt, Dec 25 2016
a(n) = A093353(n) + n + 1, conjectured. - Matej Veselovac, Jan 21 2020

A259661 Binary representation of the middle column of the "Rule 54" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 11, 110, 1100, 11001, 110011, 1100110, 11001100, 110011001, 1100110011, 11001100110, 110011001100, 1100110011001, 11001100110011, 110011001100110, 1100110011001100, 11001100110011001, 110011001100110011, 1100110011001100110, 11001100110011001100

Offset: 0

Robert Price, Dec 05 2015

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 8 rows at left, ignoring "0" outside of range of 1's, the center column values in parentheses, and at right the value of center column cells up to that row:
                (1)                 -> 1
              1 (1) 1               -> 11
            1 0 (0) 0 1             -> 110
          1 1 1 (0) 1 1 1           -> 1100
        1 0 0 0 (1) 0 0 0 1         -> 11001
      1 1 1 0 1 (1) 1 0 1 1 1       -> 110011
    1 0 0 0 1 0 (0) 0 1 0 0 0 1     -> 1100110
  1 1 1 0 1 1 1 (0) 1 1 1 0 1 1 1   -> 11001100
1 0 0 0 1 0 0 0 (1) 0 0 0 1 0 0 0 1 -> 110011001
(End)

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Rule 54
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A071030, A118109, A118108, A133872, A077854.

lim = 20; Take[Last@ Take[#, Ceiling[Length[#]/2]] & /@ CellularAutomaton[54, {{1}, 0}, lim], #] & /@ Range@ lim (* Michael De Vlieger, Dec 09 2015 *)

Conjectures from Colin Barker, Dec 08 2015 and Apr 17 2019: (Start)
a(n) = 11*a(n-1) - 11*a(n-2) + 11*a(n-3) - 10*a(n-4) for n>3.
G.f.: 1 / ((1-x)*(1-10*x)*(1+x^2)).
(End)

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A071030 Triangle read by rows giving successive states of cellular automaton generated by "Rule 54".

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A118108 Decimal representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.

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A265225 Total number of ON (black) cells after n iterations of the "Rule 54" elementary cellular automaton starting with a single ON (black) cell.

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A259661 Binary representation of the middle column of the "Rule 54" elementary cellular automaton starting with a single ON (black) cell.

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