A269160 -id:A269160 - cp's OEIS frontend

A269164 Numbers not in range of A269160; indices of zeros in A269162 from n >= 1 onward.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Antti Karttunen, Feb 20 2016

nonn

Natural numbers n for which there does not exist any number k such that A269160(k) = n.

These are binary representations (shown in decimal) of Garden of Eden patterns in Wolfram's Rule 30 cellular automaton if infinite predecessors are forbidden.

Antti Karttunen, Table of n, a(n) for n = 1..12285
Eric Weisstein's World of Mathematics, Rule 30
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Complement: A269163.
Indices of zeros in A269162 and A269166 (after the initial zero).
Cf. A269169 (left inverse).
Cf. A269160, A269168.

terms = 100; Clear[f]; f[max_] := f[max] = (s = Sort[Table[BitXor[n, BitOr[2n, 4n]], {n, 0, max}]]; Complement[Range[Last[s]], s][[1 ;; terms ]]); f[terms]; f[max = 2 terms]; While[Print[max]; f[max] != f[max/2], max = 2 max]; A269164 = f[max] (* Jean-François Alcover, Feb 23 2016 *)

Other identities. For all n >= 1:

A269169(a(n)) = n.

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).

A269163 Numbers which have a finite predecessor in Wolfram's Rule 30 cellular automaton; range of A269160 sorted into ascending order.

Original entry on oeis.org

0, 7, 13, 14, 25, 26, 27, 28, 49, 50, 51, 52, 53, 54, 56, 63, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 111, 112, 119, 125, 126, 193, 194, 195, 196, 197, 198, 200, 201, 202, 203, 204, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 221, 222, 223, 224, 231, 237, 238, 249, 250, 251, 252, 385, 386, 387, 388

Offset: 0

Antti Karttunen, Feb 20 2016

nonn

Numbers which have a finite predecessor in Wolfram's Rule 30 cellular automaton. The configuration of white and black cells is encoded in the binary representation (A007088) of each number.

The indexing starts from zero, because a(0) = 0 is a special case in this sequence. (Zero is the only number which is its own predecessor).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Eric Weisstein's World of Mathematics, Rule 30
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Complement: A269164.

Cf. A007088, A269162.

terms = 100; Clear[f]; f[max_] := f[max] = Sort[Table[BitXor[n, BitOr[2n, 4n]], {n, 0, max}]][[1 ;; terms]]; f[terms]; f[max = 2 terms]; While[ Print[max]; f[max] != f[max/2], max = 2 max]; A269163 = f[max] (* Jean-François Alcover, Feb 23 2016 *)

A269168 Rule 30 binary tree permutation: a(1) = 1, a(2n) = A269160(a(n)), a(2n+1) = A269164(1+a(n)).

Original entry on oeis.org

1, 7, 2, 25, 9, 14, 3, 111, 33, 63, 11, 50, 18, 13, 4, 401, 143, 231, 41, 193, 79, 53, 15, 222, 66, 126, 22, 51, 17, 28, 5, 1783, 529, 945, 175, 825, 295, 223, 55, 839, 257, 497, 95, 203, 69, 49, 19, 802, 286, 462, 82, 386, 158, 106, 30, 221, 67, 119, 21, 100, 36, 27, 6, 6409, 2295, 3703, 657, 3159, 1201, 849, 233

Offset: 1

Antti Karttunen, Feb 21 2016

This sequence can be represented as a binary tree. Each left hand child is produced as A269160(n), and each right hand child as A269164(1+n), when the parent node contains n:
                                     |
                  ...................1...................
                 7                                       2
      25......../ \........9                  14......../ \........3
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  111      33         63       11         50       18         13       4
401 143  231 41    193  79   53  15    222  66  126  22     51  17   28 5
etc.
Each maximal leftward branch (e.g. 1, 7, 25, ... (= A110240) or 9, 63, 193, ... or 2, 14, 50, ...) gives a trajectory of Rule 30 cellular automaton starting from a particular "seed configuration" which are given in A269164.

Antti Karttunen, Table of n, a(n) for n = 1..1023
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences that are permutations of the natural numbers

Inverse: A269167.
Cf. A269160, A269164.
Cf. A110240 (the left edge).

nmax = (* sequence length *) 100; terms (* from A269164 *) = 2000; Clear[a, f]; A269160[n_] := BitXor[n, BitOr[2 n, 4 n]]; f[max_] := f[max] = (s = Sort[Table[A269160[n], {n, 0, max}]]; Complement[Range[Last[s]], s][[1 ;; terms]]); f[terms]; f[max = 2 terms]; While[f[max] != f[max/2], max = 2 max]; A269164[n_Integer] := If[n > Length[f[max]], 0, f[max][[n]]]; a[1] = 1; a[n_] := a[n] = If[EvenQ[n], A269160[a[n/2]], A269164[1 + a[(n - 1)/2]]]; A269168 = Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Feb 23 2016 *)

a(1) = 1, after which, a(2n) = A269160(a(n)), a(2n+1) = A269164(1+a(n)).

A269167 Permutation of natural numbers: a(1) = 1, a(A269160(n)) = 2a(n), a(A269164(n+1)) = 1+(2a(n)).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 2, 127, 5, 255, 11, 511, 14, 6, 23, 1023, 29, 13, 47, 2047, 59, 27, 95, 4095, 4, 126, 62, 30, 119, 55, 191, 8191, 9, 253, 125, 61, 239, 111, 383, 16383, 19, 507, 251, 123, 479, 223, 767, 32767, 46, 12, 28, 1022, 22, 510, 39, 254, 1015, 503, 247, 959, 447, 1535, 10, 65535, 93, 25, 57, 2045, 45

Offset: 1

Antti Karttunen, Feb 21 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12285
Index entries for sequences that are permutations of the natural numbers

Inverse: A269168.

Cf. A269160, A269162, A269163, A269164, A269169.

terms = 100; A269160[n_] := BitXor[n, BitOr[2 n, 4 n]]; f[max_] := f[max] = (s = Sort[Table[A269160[n], {n, 0, max}]]; Complement[Range[Last[s]], s][[1 ;; terms]]); f[terms]; f[max = 2 terms]; While[f[max] != f[max/2], max = 2 max]; A269164[n_] := f[max][[n]]; a[1]=1; eq[n_] := a[A269160[n]] == 2*a[n] && a[A269164[n+1]] == 1 + 2*a[n]; A269167 = Array[a, terms-1] /. Solve[Array[eq, terms-1]] // First (* Jean-François Alcover, Feb 23 2016 *)

a(1) = 1, for n > 1, if A269162(n) > 0 [when n is in A269163], a(n) = 2*a(A269162(n)), otherwise [when n is in A269164], a(n) = 1 + 2*a(A269169(n)-1).

A110240 Decimal form of binary integer produced by the ON cells at n-th generation following Wolfram's Rule 30 cellular automaton starting from a single ON-cell represented as 1.

Original entry on oeis.org

1, 7, 25, 111, 401, 1783, 6409, 28479, 102849, 456263, 1641433, 7287855, 26332369, 116815671, 420186569, 1865727615, 6741246849, 29904391303, 107568396185, 477630335215, 1725755276049, 7655529137527, 27537575631497

Offset: 0

Alexandre Wajnberg and Eric Angelini, Sep 06 2005

See A245549 for binary equivalents. See A070952 for number of ON cells. - N. J. A. Sloane, Jul 28 2014
For n > 0: 3 < a(n+1) / a(n) < 5, floor(a(n+1)/a(n)) = A010702(n+1). - Reinhard Zumkeller, Jun 08 2013
Iterates of A269160 starting from a(0) = 1. See also A269168. - Antti Karttunen, Feb 20 2016
Also, the decimal representation of the n-th generation of the "Rule 66847740" 5-neighbors elementary cellular automaton starting with a single ON (black) cell. - Philipp O. Tsvetkov, Jul 17 2019

			a(1)=1 because the automaton begins at first "generation" with one black cell: 1;
a(2)=5 because one black cell, through Rule 30 at 2nd generation, produces three contiguous black cells: 111 (binary), so 7 (decimal);
a(3)=25 because the third generation is "black black white white black" cells: 11001, so 25 (decimal).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Erica Jen, Global properties of cellular automata, Journal of Statistical Physics 43 (1986), pp. 219-242.
Antti Karttunen, Terms up to a(255) drawn as binary strings, with 1 bit = 3x3 pixels resolution
Antti Karttunen, Terms up to a(1023) drawn as binary strings, with 1 bit = 1 pixel resolution
N. J. A. Sloane, Illustration of first 20 generations
Eric Weisstein's World of Mathematics, Rule 30.
Stephen Wolfram, Announcing the Rule 30 Prizes, 2019
Index entries for sequences related to cellular automata

Cf. A030101, A070950, A051023, A092539, A092540, A070952 (number of ON cells, the binary weight of terms), A100053, A100054, A100055, A094603, A094604, A000225, A074890, A010702, A245549, A269160, A269162.
Cf. A269165 (indices of ones in this sequence).
Cf. A269166 (a left inverse).
Left edge of A269168.
Cf. also A265281, A328106.
For bitwise XOR (and OR) combinations with other such 1D CA trajectories, see for example: A327971, A327972, A327973, A327976, A328103, A328104.

a110240 = foldl (\v d -> 2 * v + d) 0 . map toInteger . a070950_row
-- Reinhard Zumkeller, Jun 08 2013

rows = 23; ca = CellularAutomaton[30, {{1}, 0}, rows-1]; Table[ FromDigits[ ca[[k, rows-k+1 ;; rows+k-1]], 2], {k, 1, rows}] (* Jean-François Alcover, Jun 07 2012 *)

A269160(n) = bitxor(n, bitor(2*n, 4*n));
A110240(n) = if(!n,1,A269160(A110240(n-1))); \\ Antti Karttunen, Oct 05 2019

def A269160(n): return(n^((n<<1)|(n<<2)))
def genA110240():
    '''Yield successive terms of A110240 (Rule 30) starting from A110240(0)=1.'''
    s = 1
    while True:
       yield s
       s = A269160(s)
def take(n, g):
    '''Returns a list composed of the next n elements returned by generator g.'''
    z = []
    if 0 == n: return(z)
    for x in g:
        z.append(x)
        if n > 1: n = n-1
        else: return(z)
take(30, genA110240())
# Antti Karttunen, Oct 05 2019

;; With memoization-macro definec.
(definec (A110240 n) (if (zero? n) 1 (A269160 (A110240 (- n 1)))))
;; Antti Karttunen, Feb 20 2016

From Antti Karttunen, Feb 20 2016: (Start)
a(0) = 1, for n >= 1, a(n) = A269160(a(n-1)).
a(n) = A030101(A265281(n)). [The rule 30 is the mirror image of the rule 86.]
A269166(a(n)) = n for all n >= 0. (End)
From Antti Karttunen, Oct 05 2019: (Start)
For n >= 1, a(n) = a(n-1) XOR 2*A328104(n-1).
For n >= 1, a(n) = 2*a(n-1) XOR A327973(n). (End)

More terms from Eric W. Weisstein, Apr 08 2006

Offset corrected by Reinhard Zumkeller, Jun 08 2013

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

A327973 Bitwise XOR of two successive generations (centrally aligned) in the trajectory of rule 30 started from a lone 1 cell: a(n) = A110240(n) XOR 2*A110240(n-1).

Original entry on oeis.org

5, 23, 93, 335, 1493, 5351, 23853, 85951, 382405, 1369943, 6103965, 21996687, 97906325, 350709671, 1562619373, 5631262591, 25064000389, 89782414999, 400033474525, 1441615751887, 6416397448021, 22984338788455, 102408232210605, 369052763468095, 1642598765228869, 5883986891577303, 26216498605021469, 94477513773305103

Offset: 1

Antti Karttunen, Oct 03 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Antti Karttunen, Terms a(1)-a(256) drawn as binary strings, with 1 bit = 3x3 pixels resolution
Antti Karttunen, Terms a(1)-a(1024) drawn as binary strings, with 1 bit = 1 pixel resolution
Index entries for sequences related to cellular automata

Cf. A110240, A269160, A327974 (gives the middle bit), A328107 (binary weight of terms).

Cf. also A327971, A327972, A327976, A328103, A328104 for other such combinations.

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));
\\ Use this one for writing b-files:
A327973write(up_to) = { my(s=1, t, n=0); for(n=1,up_to, t = A269160(s); write("b327973.txt", n, " ", bitxor(2*s, t)); s = t); };

def A269160(n): return(n^((n<<1)|(n<<2)))
def genA327973():
    '''Yield successive terms of A327973.'''
    s = 1
    while True:
       t = A269160(s)
       yield (t^(s<<1))
       s = t

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

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 *)

cp's OEIS Frontend

A269164 Numbers not in range of A269160; indices of zeros in A269162 from n >= 1 onward.

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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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A269163 Numbers which have a finite predecessor in Wolfram's Rule 30 cellular automaton; range of A269160 sorted into ascending order.

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A269168 Rule 30 binary tree permutation: a(1) = 1, a(2n) = A269160(a(n)), a(2n+1) = A269164(1+a(n)).

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A269167 Permutation of natural numbers: a(1) = 1, a(A269160(n)) = 2*a(n), a(A269164(n+1)) = 1+(2*a(n)).

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A110240 Decimal form of binary integer produced by the ON cells at n-th generation following Wolfram's Rule 30 cellular automaton starting from a single ON-cell represented as 1.

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

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Extensions

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

A269167 Permutation of natural numbers: a(1) = 1, a(A269160(n)) = 2a(n), a(A269164(n+1)) = 1+(2a(n)).