A071030 -id:A071030 - cp's OEIS frontend

A071045 Number of 0's in n-th row of triangle in A071030.

0, 0, 3, 1, 6, 2, 9, 3, 12, 4, 15, 5, 18, 6, 21, 7, 24, 8, 27, 9, 30, 10, 33, 11, 36, 12, 39, 13, 42, 14, 45, 15, 48, 16, 51, 17, 54, 18, 57, 19, 60, 20, 63, 21, 66, 22, 69, 23, 72, 24, 75, 25, 78, 26, 81, 27, 84, 28, 87, 29, 90, 30, 93, 31, 96, 32, 99, 33, 102, 34

Offset: 0

Hans Havermann, May 26 2002

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

Robert Price, Table of n, a(n) for n = 0..999
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A001057, A064455, A071030.

a := n -> n + ((-1)^n*(2*n + 1) - 1)/4;
seq(a(n), n=0..69); # Peter Luschny, Feb 11 2019

LinearRecurrence[{0, 2, 0, -1}, {0, 0, 3, 1}, 70] (* Jean-François Alcover, Jul 08 2019 *)

a(n) = Sum_{k=0..n-1} Sum_{i=0..k} C(i,k) - (-1)^k. - Wesley Ivan Hurt, Sep 20 2017
a(n) = n + ((-1)^n*(2*n + 1) - 1)/4 = n - A001057(n). - Peter Luschny, Feb 11 2019
For n > 0: a(n) = (n^2 - 1) mod (2*n + 1). - Ctibor O. Zizka, Mar 11 2025

A064455 a(2n) = 3n, a(2n-1) = n.

Original entry on oeis.org

1, 3, 2, 6, 3, 9, 4, 12, 5, 15, 6, 18, 7, 21, 8, 24, 9, 27, 10, 30, 11, 33, 12, 36, 13, 39, 14, 42, 15, 45, 16, 48, 17, 51, 18, 54, 19, 57, 20, 60, 21, 63, 22, 66, 23, 69, 24, 72, 25, 75, 26, 78, 27, 81, 28, 84, 29, 87, 30, 90, 31, 93, 32, 96, 33, 99, 34, 102, 35, 105, 36, 108

Offset: 1

N. J. A. Sloane, Oct 02 2001

Also number of 1's in n-th row of triangle in A071030. - Hans Havermann, May 26 2002
Number of ON cells at generation n of 1-D CA defined by Rule 54. - N. J. A. Sloane, Aug 09 2014
a(n)*A098557(n) equals the second right hand column of A167556. - Johannes W. Meijer, Nov 12 2009
Given a(1) = 1, for all n > 1, a(n) is the least positive integer not equal to a(n-1) such that the arithmetic mean of the first n terms is an integer. The sequence of arithmetic means of the first 1, 2, 3, ..., terms is 1, 2, 2, 3, 3, 4, 4, ... (A004526 disregarding its first three terms). - Rick L. Shepherd, Aug 20 2013

			a(13) = a(2*7 - 1) = 7, a(14) = a(2*7) = 21.
a(8) = 8-9+10-11+12-13+14-15+16 = 12. - _Bruno Berselli_, Jun 05 2013

Harry J. Smith, Table of n, a(n) for n = 1..1000
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, 2016.
S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601--644.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)

Interleaving of A000027 and A008585 (without first term).

Cf. A004526, A052928, A064433, A071030, A080512, A098557, A123684, A137501, A167556, A225144, A265888.

maxarg := 75; for n := 1 to maxarg do if n mod 2 = 1 then write((n+1) div 2, " ") else write((n div 2)*3," "); end; end;

a:=[];;  for n in [1..75] do if n mod 2 = 0 then Add(a,3*n/2); else Add(a,(n+1)/2); fi; od; a; # Muniru A Asiru, Oct 28 2018

import Data.List (transpose)
a064455 n = n + if m == 0 then n' else - n'  where (n',m) = divMod n 2
a064455_list = concat $ transpose [[1 ..], [3, 6 ..]]
-- Reinhard Zumkeller, Oct 12 2013

[(1/2)*n*(-1)^n+n+(1/4)*(1-(-1)^n): n in [1..80]]; // Vincenzo Librandi, Aug 10 2014

A064455 := proc(n)
    if type(n,'even') then
        3*n/2 ;
    else
        (n+1)/2 ;
    end if;
end proc: # R. J. Mathar, Aug 03 2015

Table[ If[ EvenQ[n], 3n/2, (n + 1)/2], {n, 1, 70} ]

a(n) = { if (n%2, (n + 1)/2, 3*n/2) } \\ Harry J. Smith, Sep 14 2009

a(n)=if(n<3,2*n-1,((n-1)*(n-2))%(2*n-1)) \\ Jim Singh, Oct 14 2018

def A064455(n): return (3*n - (2*n-1)*(n%2))//2
print([A064455(n) for n in range(1,81)]) # G. C. Greubel, Jan 30 2025

a(n) = (1/2)*n*(-1)^n + n + (1/4)*(1 - (-1)^n). - Stephen Crowley, Aug 10 2009
G.f.: x*(1+3*x) / ( (1-x)^2*(1+x)^2 ). - R. J. Mathar, Mar 30 2011
From Jaroslav Krizek, Mar 22 2011: (Start)
a(n) = n - A123684(n-1) for odd n.
a(n) = n + a(n-1) for even n.
a(n) = A123684(n) + A137501(n).
Abs( a(n) - A123684(n) ) =  A052928(n). (End)
a(n) = Sum_{i=n..2*n} i*(-1)^i. - Bruno Berselli, Jun 05 2013
a(n) = n + floor(n/2)*(-1)^(n mod 2). - Bruno Berselli, Dec 14 2015
a(n) = (n^2-3*n+2) mod (2*n-1) for n>2. - Jim Singh, Oct 31 2018
E.g.f.: (1/2)*(x*cosh(x) + (1+3*x)*sinh(x)). - G. C. Greubel, Jan 30 2025

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

A118109 Binary representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.

Original entry on oeis.org

1, 111, 10001, 1110111, 100010001, 11101110111, 1000100010001, 111011101110111, 10001000100010001, 1110111011101110111, 100010001000100010001, 11101110111011101110111, 1000100010001000100010001, 111011101110111011101110111, 10001000100010001000100010001, 1110111011101110111011101110111

Offset: 0

Eric W. Weisstein, Apr 13 2006

			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 (A118108):
                   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
10001 0001 0001 0001 = 69905
(End)

Robert Price, Table of n, a(n) for n = 0..499
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 (essentially the same but lists bits separately), A118108 (converted to base 10).

rule=54; 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 *) Table[FromDigits[catri[[k]]], {k, 1, rows}] (* Binary Representation of Rows *) (* Robert Price, Feb 21 2016 *)

Conjectures from Colin Barker, Dec 08 2015 and Apr 16 2019: (Start)
a(n) = 10001*a(n-2)-10000*a(n-4) for n>3.
G.f.: (1+111*x) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)).
(End)
Conjecture: a(n) = floor((10000+1100*(n mod 2))*100^n/9999). - Karl V. Keller, Jr., Sep 24 2021

Terms changed to match definition, as suggested by Michael De Vlieger. - N. J. A. Sloane, Oct 17 2015

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)

A204714 Maximum period of cellular automaton rule 54 in a cyclic universe of width n.

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 4, 8, 27, 30, 99, 12, 169, 112, 330, 40, 289, 306, 494, 86, 399, 484, 690, 312, 1800, 624, 918, 224, 783, 780, 1240, 608, 1056, 952, 1540, 684

Offset: 1

Ben Branman, Jan 18 2012

nonn

			For n=8, the initial condition 00011101 yields the evolution
00011101
10100011
01110100
10001110
11010001
00111010
01000111
11101000
00011101
Which is period 8, the maximum possible, so a(8)=8.

Index entries for sequences related to cellular automata
Eric Weisstein's World of Mathematics, Rule 54

Cf. A071030, A118108

f[list_] := -Subtract @@ Flatten[Map[Position[#, #[[-1]]] &, NestWhileList[CellularAutomaton[54], list, Unequal, All], {0}]]; a[n_] := Max[Table[f[IntegerDigits[i, 2, n]], {i, 0, 2^n - 1}]]; Table[a[n], {n, 1, 10}]

a(15)-a(36) from Lars Blomberg, Dec 24 2015

cp's OEIS Frontend

A071045 Number of 0's in n-th row of triangle in A071030.

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A064455 a(2n) = 3n, a(2n-1) = n.

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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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A118109 Binary representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single 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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A204714 Maximum period of cellular automaton rule 54 in a cyclic universe of width n.

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