A110240 -id:A110240 - cp's OEIS frontend

A070950 Triangle read by rows giving successive states of cellular automaton generated by "Rule 30".

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

Offset: 0

N. J. A. Sloane, May 19 2002

If cell and right-hand neighbor are both 0 then new state of cell = state of left-hand neighbor; otherwise new state is complement of that of left-hand neighbor.
A simple rule which produces apparently random behavior. "... probably the single most surprising discovery I have ever made" - Stephen Wolfram.
Row n has length 2n+1.
A110240(n) = A245549(n) = value of row n, seen as binary number. - Reinhard Zumkeller, Jun 08 2013
A070952 gives number of ON cells. - N. J. A. Sloane, Jul 28 2014

			Triangle begins:
        1;
      1,1,1;
    1,1,0,0,1;
  1,1,0,1,1,1,1;
  ...

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 27.

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
N. J. A. Sloane, Illustration of initial terms
Eric Weisstein's World of Mathematics, Rule 30
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

Cf. A070951, A070952 (row sums), A051023 (central terms).
Cf. A071032 (mirror image, rule 86), A226463 (complemented, rule 135), A226464 (mirrored and complemented, rule 149).
Cf. A363343 (diagonals from the right), A363344 (diagonals from the left).
Cf. A094605 (periods of diagonals from the right), A363345 (eventual periods of diagonals from the left), A363346 (length of initial transients on diagonals from the left).
Cf. also A245549, A110240.

a070950 n k = a070950_tabf !! n !! k
a070950_row n = a070950_tabf !! n
a070950_tabf = iterate rule30 [1] where
   rule30 row = f ([0,0] ++ row ++ [0,0]) where
       f [,]          = []
       f (u:ws@(0:0:_)) = u : f ws
       f (u:ws)         = (1 - u) : f ws
-- Reinhard Zumkeller, Feb 01 2013

ArrayPlot[CellularAutomaton[30,{{1},0}, 50]] (* N. J. A. Sloane, Aug 11 2009 *)
Clear[t, n, k]; nn = 10; t[1, k_] := t[1, k] = If[k == 3, 1, 0];
t[n_, k_] := t[n, k] = Mod[t[-1 + n, -2 + k] + t[-1 + n, -1 + k] + (1 + t[-1 + n, -1 + k]) t[-1 + n, k], 2]; Flatten[Table[Table[t[n, k], {k, 3, 2*n + 1}], {n, 1, nn}]] (*Mats Granvik,Dec 08 2019*)

From Mats Granvik, Dec 06 2019: (Start)
The following recurrence expresses the rules in rule 30, except that instead of If, Or, And, Not, we use addition, subtraction, and multiplication.
T(n, 1) = 0
T(n, 2) = 0
T(1, 3) = 1
T(n, k) = [2*n + 1 >= k] ((1 - (T(n - 1, k - 2)*T(n - 1, k - 1)*T(n - 1, k)))*(1 - T(n - 1, k - 2)*T(n - 1, k - 1)*(1 - T(n - 1, k)))*(1 - (T(n - 1, k - 2)*(1 - T(n - 1, k - 1))*T(n - 1, k)))*(1 - ((1 - T(n - 1, k - 2))*(1 - T(n - 1, k - 1))*(1 - T(n - 1, k))))) + ((T(n - 1, k - 2)*(1 - T(n - 1, k - 1))*(1 - T(n - 1, k)))*((1 - T(n - 1, k - 2))*T(n - 1, k - 1)*T(n - 1, k))*((1 - T(n - 1, k - 2))*T(n - 1, k - 1)*(1 - T(n - 1, k)))*((1 - T(n - 1, k - 2))*(1 - T(n - 1, k - 1))*T(n - 1, k))).
Discarding the term after the plus sign, multiplying/expanding the terms out and replacing all exponents with ones, gives us this simplified recurrence:
T(n, 1) = 0
T(n, 2) = 0
T(1, 3) = 1
T(n, k) = T(-1 + n, -2 + k) + T(-1 + n, -1 + k) - 2 T(-1 + n, -2 + k) T(-1 + n, -1 + k) + (-1 + 2 T(-1 + n, -2 + k)) (-1 + T(-1 + n, -1 + k)) T(-1 + n, k).
That in turn simplifies to:
T(n, 1) = 0
T(n, 2) = 0
T(1, 3) = 1
T(n, k) = Mod(T(-1 + n, -2 + k) + T(-1 + n, -1 + k) + (1 + T(-1 + n, -1 + k)) T(-1 + n, k), 2).
(End)

More terms from Hans Havermann, May 24 2002

A051023 Middle column of rule-30 1-D cellular automaton, from a lone 1 cell.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein

nonn

A092539(n) gives the value of prefix of length n+1, seen as a binary number. - Reinhard Zumkeller, Jun 08 2013

Also middle column of rule 86 1-D cellular automaton, from a lone 1 cell, as rule 86 is the mirror image of rule 30. - Antti Karttunen, Oct 03 2019

Antti Karttunen, Table of n, a(n) for n = 0..100000 (terms 0..10000 from Reinhard Zumkeller)
Pedro Hecht, PQC: R-Propping a Chaotic Cellular Automata, Univ. of Buenos Aires (Argentina, 2021).
Erica Jen, Global properties of cellular automata, Journal of Statistical Physics 43 (1986), pp 219-242.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Eric Weisstein's World of Mathematics, Rule 30
Eric Weisstein's World of Mathematics, Random Number
Xiangdong Wen, A Billion Bits of the Center Column of the Rule 30 Cellular Automaton, 2019
Stephen Wolfram, Announcing the Rule 30 Prizes, 2019
Stephen Wolfram, A Million Bits of the Center Column of the Rule 30 Cellular Automaton, 2017
Wolfram Mathematica, Cellular automata: Mathematica notebooks
Index entries for sequences related to cellular automata
Index entries for characteristic functions

Cf. A070950, A269160 (rule 30), A071032, A269161 (rule 86).
Cf. A000035, A092539, A226474.
Cf. A327974 (adjacent bits xored), A327982 (partial sums), A327983 (run lengths).
Characteristic function of A327984 (gives the positions of ones in this sequence), A327985 (positions of zeros).
Cf. also A328100, A328101, A328102 (neighbor columns).
Cf. A365254 (converted to base 10).

a051023 n = a070950 n n  -- Reinhard Zumkeller, Jun 06 2013

CellularAutomaton[30, {{1}, 0}, 101, {All, {0}}]//Flatten

A051023(n) = ((A110240(n)>>n)%2);
\\ Or for fast creation of b-files:
A051023write(up_to) = { my(s=1, n=0); for(n=0,up_to, write("b051023.txt", n, " ", ((s>>n)%2)); s = A269160(s)); }; \\ Antti Karttunen, Oct 03 2019

a(n) = A070950(n,n). - Reinhard Zumkeller, Jun 06 2013
a(n) = 1 - A226474(n). - Reinhard Zumkeller, Jun 08 2013
From Antti Karttunen, Oct 04 2019: (Start)
a(n) = A000035(floor(A110240(n) / 2^n)).
For n>= 2, a(n) = (A328100(n) OR A328101(n)) XOR A328101(1+n). ["Sideways evaluation"]
(End)

Corrected from 64th term by Daniel B. Cristofani (cristofd(AT)hevanet.com), Jan 07 2004

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

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

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

			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, ...

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Illustration of first 20 generations
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Rule 30
Wikipedia, Rule 30
Index entries for sequences related to cellular automata

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.

a070952 = sum . a070950_row  -- Reinhard Zumkeller, Jun 07 2013

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 *)

More terms from Hans Havermann, May 26 2002

Corrected offset and initial term - N. J. A. Sloane, Jun 07 2013

A245549 State of one-dimensional cellular automaton 'sigma' (Rule 30): 000,001,010,011,100,101,110,111 -> 0,0,0,1,1,1,1,0 at generation n, regarded as a binary number.

Original entry on oeis.org

1, 111, 11001, 1101111, 110010001, 11011110111, 1100100001001, 110111100111111, 11001000111000001, 1101111011001000111, 110010000101111011001, 11011110011010000101111

Offset: 0

N. J. A. Sloane, Jul 28 2014

nonn

See A110240 for decimal equivalents. See A070952 for number of ON cells.

Robert Price, Table of n, a(n) for n = 0..999
N. J. A. Sloane, Illustration of first 20 generations
Eric Weisstein's World of Mathematics, Rule 150
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A070952, A110240, A070950, A074890, A269160.

rule=30; 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 *)

A265281 Decimal representation of the n-th iteration of the "Rule 86" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 7, 19, 123, 275, 1915, 4627, 32379, 67347, 463739, 1276435, 8001147, 18245395, 124220283, 309475859, 2134808187, 4354074387, 30225757051, 82642024979, 530573137531, 1175676118803, 8199373878139, 19977819994643, 139441424012923, 286691477808915

Offset: 0

Robert Price, Dec 06 2015

Iterates of A269161 starting from a(0) = 1. - Antti Karttunen, Feb 20 2016

Also, the decimal representation of the n-th generation of the "Rule 859583292" 5-neighbors elementary cellular automaton starting with a single ON (black) cell. - Philipp O. Tsvetkov, Jul 17 2019

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

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A030101, A071032, A265280, A269161.

Cf. also A110240.

rule = 86; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]],2], {k,1,rows}]

From Antti Karttunen, Feb 20 2016: (Start)
a(0) = 1, for n >= 1, a(n) = A269161(a(n-1)).
a(n) = A030101(A110240(n)). [The rule 86 is the mirror image of the rule 30.]
(End)

A269162 a(0) = 0, for n > 0, a(n) = the least (necessarily also unique) k such that A269160(k) = n, or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 6, 5, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 14, 13, 12, 11, 10, 0, 8, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 0, 0, 25

Offset: 0

Antti Karttunen, Feb 20 2016

nonn

If n > 0 and a(n) > 0 then a(n) is the unique finite predecessor of the configuration encoded in the binary representation of n (A007088) when Wolfram's Rule 30 cellular automaton is applied.

Antti Karttunen, Table of n, a(n) for n = 0..16387
Eric Weisstein's World of Mathematics, Rule 30
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A110240, A269160, A269163, A269164 (indices of zeros), A269165, A269166.

(* empirical *) a[n_] := Module[{k}, For[k = Floor[n/7], k <= Ceiling[n/3], k++, If[BitXor[k, BitOr[2k, 4k]] == n, Return[k]]]; 0]; Table[a[n], {n, 0, 16387}] (* Jean-François Alcover, Feb 23 2016 *)

(define (A269162 n) (let loop ((p 0)) (cond ((= n (A269160 p)) p) ((> p n) 0) (else (loop (+ 1 p)))))) ;; Very slow implementation.
(define (A269162 n) (if (zero? n) n (let ((nwid-2 (- (A000523 n) 2))) (let loop ((p (if (< n 4) 0 (A000079 nwid-2)))) (let ((k (A269160 p))) (cond ((= n k) p) ((> (A000523 p) nwid-2) 0) (else (loop (+ 1 p))))))))) ;; Somewhat optimized.

Other identities. For all n >= 0:
a(A269160(n)) = n. [This sequence works as a left inverse of A269160.]
a(A110240(n+1)) = A110240(n).

A327980 Distances between successive zeros in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

Original entry on oeis.org

4, 1, 3, 1, 1, 2, 3, 1, 2, 1, 4, 2, 4, 1, 4, 2, 2, 3, 1, 1, 1, 3, 1, 2, 2, 3, 2, 2, 7, 1, 1, 1, 5, 1, 1, 2, 2, 4, 1, 1, 1, 1, 2, 1, 2, 3, 1, 1, 4, 1, 1, 3, 3, 3, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 6, 4, 2, 1, 4, 1, 1, 4, 2, 4, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 1, 1, 8, 3, 1, 2, 3, 4, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Oct 03 2019

nonn

First differences of A327985, which gives indices of zeros in A051023.

			The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:
   0:              (1)
   1:             1(1)1
   2:            11(0)01
   3:           110(1)111
   4:          1100(1)0001
   5:         11011(1)10111
   6:        110010(0)001001
   7:       1101111(0)0111111
   8:      11001000(1)11000001
   9:     110111101(1)001000111
  10:    1100100001(0)1111011001
  11:   11011110011(0)10000101111
  12:  110010001110(0)110011010001
When noting up the distances between successive 0's in its central column (indicated here with parentheses), we obtain 6-2 (as the first 0 is on row 2, and the second is on row 6), 7-6, 10-7, 11-10, 12-11, ..., that is, the first terms of this sequence: 4, 1, 3, 1, 1, ...

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to cellular automata

Cf. A051023, A110240, A245549, A269160, A327981, A327983, A327985.

A327980list[upto_]:=Differences[Flatten[Position[CellularAutomaton[30,{{1},0},{upto,{{0}}}],0]]];A327980list[300] (* Paolo Xausa, Jun 01 2023 *)

up_to = 105;
A269160(n) = bitxor(n, bitor(2*n, 4*n));
A327980list(up_to) = { my(v=vector(up_to), s=25, n=2, on=n, k=0); while(kA269160(s); if(!((s>>n)%2), k++; v[k] = (n-on); on=n)); (v); }
v327980 = A327980list(up_to);
A327980(n) = v327980[n];

a(n) = A327985(1+n) - A327985(n).

A327981 Distances between successive ones in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 4, 2, 1, 3, 3, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 2, 1, 5, 1, 3, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 4, 2, 2, 1, 1, 6, 3, 2, 1, 4, 1, 1, 4, 1, 2, 1, 2, 1, 2, 8, 4, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 1, 1, 2, 2, 1, 1, 6, 1, 3, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 1, 1

Offset: 1

Antti Karttunen, Oct 03 2019

nonn

First differences of A327984, which gives indices of ones in A051023.

			The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:
   0:              (1)
   1:             1(1)1
   2:            11(0)01
   3:           110(1)111
   4:          1100(1)0001
   5:         11011(1)10111
   6:        110010(0)001001
   7:       1101111(0)0111111
   8:      11001000(1)11000001
   9:     110111101(1)001000111
  10:    1100100001(0)1111011001
  11:   11011110011(0)10000101111
  12:  110010001110(0)110011010001
  13: 1101111011001(1)1011100110111
The distances between successive 1's in its central column (indicated here with parentheses) are 1-0 (as the first 1 is on row 0, and the second is on row 1), 3-1, 4-3, 5-4, 8-5, 9-8, 13-9, ..., that is, the first terms of this sequence: 1, 2, 1, 1, 3, 1, 4, ...

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A051023, A110240, A269160, A327980, A327982, A327983, A327984.

A327981list[upto_]:=Differences[Flatten[Position[CellularAutomaton[30,{{1},0},{upto,{{0}}}],1]]];A327981list[300] (* Paolo Xausa, Jun 27 2023 *)

up_to = 105;
A269160(n) = bitxor(n, bitor(2*n, 4*n));
A327981list(up_to) = { my(v=vector(up_to), s=1, n=0, on=n, k=0); while(kA269160(s); if((s>>n)%2, k++; v[k] = (n-on); on=n)); (v); }
v327981 = A327981list(up_to);
A327981(n) = v327981[n];

a(n) = A327984(1+n) - A327984(n).

A327984 Positions of ones in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

Original entry on oeis.org

0, 1, 3, 4, 5, 8, 9, 13, 15, 16, 19, 22, 23, 24, 26, 28, 29, 30, 33, 34, 35, 37, 39, 41, 42, 47, 48, 51, 53, 55, 56, 58, 60, 62, 63, 64, 65, 66, 67, 72, 73, 74, 75, 79, 81, 83, 84, 85, 91, 94, 96, 97, 101, 102, 103, 107, 108, 110, 111, 113, 114, 116, 124, 128, 129, 130, 131, 132, 134, 135, 136, 138, 141, 142, 143, 147

Offset: 1

Antti Karttunen, Oct 03 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to cellular automata

Cf. A051023 (characteristic function), A110240, A269160.

Cf. A327981 (first differences), A327982 (a left inverse), A327985 (complement).

A327984list[upto_]:=Flatten[Position[CellularAutomaton[30,{{1},0},{upto,{{0}}}],1]]-1;A327984list[200] (* Paolo Xausa, Jun 28 2023 *)

A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A051023(n) = ((A110240(n)>>n)%2);
isA327984(n) = A051023(n);

A327984write(up_to) = { my(s=1, n=0, k=0); while(k>n)%2, k++; write("b327984.txt", k, " ", n)); s = A269160(s); n++); (n); }; \\ For fast creation of b-files.

For all n >= 1, A327982(a(n)) = n.

A328107 Binary weight of A327973.

Original entry on oeis.org

2, 4, 5, 6, 7, 8, 9, 13, 11, 13, 13, 14, 17, 18, 19, 23, 20, 23, 24, 27, 26, 24, 23, 30, 31, 29, 29, 31, 36, 35, 36, 37, 35, 34, 35, 42, 40, 46, 41, 50, 54, 48, 52, 47, 47, 53, 47, 51, 51, 54, 48, 51, 60, 55, 56, 64, 61, 60, 59, 68, 71, 67, 65, 78, 64, 63, 68, 72, 70, 74, 79, 89, 85, 77, 85, 76, 79, 83, 78, 90, 97, 82, 87, 81

Offset: 1

Antti Karttunen, Oct 05 2019

nonn

Cf. A000120, A110240, A269160, A327973.

Cf. also A070952, A328105, A328106, A328108, A328109.

A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A327973(n) = bitxor(A110240(n), 2*A110240(n-1));
A328107(n) = hammingweight(bitxor(A110240(n), 2*A110240(n-1)));
\\ Use this one for writing b-files:
A328107write(up_to) = { my(s=1, t, n=0); for(n=1,up_to, t = A269160(s); write("b328107.txt", n, " ", hammingweight(bitxor(2*s, t))); s = t); };

a(n) = A000120(A327973(n)) = A000120(A110240(n) XOR 2*A110240(n-1)).

cp's OEIS Frontend

A070950 Triangle read by rows giving successive states of cellular automaton generated by "Rule 30".

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A051023 Middle column of rule-30 1-D cellular automaton, from a lone 1 cell.

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

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A245549 State of one-dimensional cellular automaton 'sigma' (Rule 30): 000,001,010,011,100,101,110,111 -> 0,0,0,1,1,1,1,0 at generation n, regarded as a binary number.

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A265281 Decimal representation of the n-th iteration of the "Rule 86" elementary cellular automaton starting with a single ON (black) cell.

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A269162 a(0) = 0, for n > 0, a(n) = the least (necessarily also unique) k such that A269160(k) = n, or 0 if no such k exists.

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A327980 Distances between successive zeros in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

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