A094604 -id:A094604 - cp's OEIS frontend

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.

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

A094605 a(n) is the period of the n-th diagonal, from the right, of Rule 30 (begun from an initial black cell).

Original entry on oeis.org

1, 2, 2, 4, 8, 8, 16, 32, 32, 64, 64, 64, 64, 64, 64, 128, 256, 256, 256, 256, 256, 256, 256, 256, 512, 1024, 1024, 2048, 2048, 4096, 4096, 4096, 4096, 4096, 8192, 8192, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 262144, 262144

Offset: 1

Eric Rowland, May 13 2004

nonn

One 64 term is missing in Wolfram 2002 (p. 871). (There are six.)

Stephen Wolfram, A New Kind of Science. Wolfram Media, 2002, p. 871.

Eric Rowland, Local nested structure in rule 30, Complex Systems 16 (2006) 239-258.
Stephen Wolfram, [Page 871] A New Kind of Science | Online.

Cf. A094604, A094606, A364773.

A094603 a(n) is the length of the maximal sequence of rightmost black cells in the n-th row of Rule 30 (begun from an initial black cell).

Original entry on oeis.org

1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 7, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 9, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 7, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 15, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 7, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 9, 1, 3, 1, 4, 1, 3, 1, 6, 1

Offset: 1

Eric Rowland, May 13 2004

New numbers (A094604) in the sequence occur at positions 2^m.

			From _Michael De Vlieger_, Oct 06 2015: (Start)
First 12 rows, replacing "0" with ".", ignoring "0" outside of range of 1's, for better visibility of ON cells, the number of contiguous rightmost black cells in each row appears at left:
1                          1
3                        1 1 1
1                      1 1 . . 1
4                    1 1 . 1 1 1 1
1                  1 1 . . 1 . . . 1
3                1 1 . 1 1 1 1 . 1 1 1
1              1 1 . . 1 . . . . 1 . . 1
6            1 1 . 1 1 1 1 . . 1 1 1 1 1 1
1          1 1 . . 1 . . . 1 1 1 . . . . . 1
3        1 1 . 1 1 1 1 . 1 1 . . 1 . . . 1 1 1
1      1 1 . . 1 . . . . 1 . 1 1 1 1 . 1 1 . . 1
4    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)

Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Rowland, Local nested structure in rule 30, Complex Systems 16 (2006) 239-258.
Eric Weisstein's World of Mathematics, Rule 30
Index entries for sequences related to cellular automata

Cf. A007814, A094604.

rows = 105; ca = CellularAutomaton[30, {{1}, 0}, rows-1]; a[n_] := Length[ Last[ Split[ ca[[n, rows-n+1 ;; rows+n-1]]]]]; Table[ a[n], {n, 1, rows}] (* Jean-François Alcover, Oct 11 2012 *)
Length /@ Map[Last, Split /@ CellularAutomaton[30, {{1}, 0}, 120] /. 0 -> Nothing /. {} -> Nothing] (* Michael De Vlieger, Oct 06 2015 *)

Conjecture: a(n) = A094604(A007814(n)). - Alan Michael Gómez Calderón, Feb 17 2023

A110267 Total number of black cells at the first n generations of a single black cell following Wolfram's Rule 30 cellular automaton.

Original entry on oeis.org

1, 4, 7, 13, 17, 26, 31, 43, 50, 62, 73, 87, 99, 118, 131, 153, 168, 187, 207, 231, 252, 275, 298, 326, 352, 379, 405, 438, 468, 502, 533, 572, 598, 637, 666, 712, 744, 788, 826, 871, 918, 959, 1004, 1053, 1091, 1146, 1188, 1239, 1283, 1336, 1379, 1438, 1490

Offset: 0

Alexandre Wajnberg, Sep 06 2005

At each generation, "looking back", one can see "behind", groups of black cells: total number of black cells (cumulative sum of first n terms of A070952).

			a(1)=1 because one black cell;
a(2)=4 because there are now 3 contiguous black cell connected to the first one, which form one only black surface of 4 cells;
a(3)=7 because appear three black cells: 4+3=7
From _Michael De Vlieger_, Dec 16 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of ON cells per row, and the running total up to that row:
                1                  =  1 ->   1
              1 1 1                =  3 ->   4
            1 1 . . 1              =  3 ->   7
          1 1 . 1 1 1 1            =  6 ->  13
        1 1 . . 1 . . . 1          =  4 ->  17
      1 1 . 1 1 1 1 . 1 1 1        =  9 ->  26
    1 1 . . 1 . . . . 1 . . 1      =  5 ->  31
  1 1 . 1 1 1 1 . . 1 1 1 1 1 1    = 12 ->  43
1 1 . . 1 . . . 1 1 1 . . . . . 1  =  7 ->  50
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Rule 30.
Index entries for sequences related to cellular automata

Cf. A070950, A051023, A092539, A092540, A070952, A100053, A100054, A100055, A094603, A094604, A000225, A074890.

See A265704 for an essentially identical sequence.

a110267 n = a110267_list !! (n-1)
a110267_list = scanl1 (+) a070952_list
-- Reinhard Zumkeller, Jun 08 2013

Accumulate[Total /@ CellularAutomaton[30, {{1}, 0}, 52]] (* Michael De Vlieger, Dec 16 2015 *)

Offset changed by Reinhard Zumkeller, Jun 08 2013

A094606 Base-2 logarithm of A094605.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 11, 11, 12, 12, 12, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 18, 18, 18, 18, 19, 20, 20, 21, 21, 21, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 27, 27, 28, 28, 28, 29

Offset: 1

Eric Rowland, May 13 2004

nonn

The positive integer n appears a(n)-a(n-1) times, where a(n) is A094604.

Eric Rowland, Local nested structure in rule 30, Complex Systems 16 (2006) 239-258.

Cf. A094604, A094605.

A110266 Number of blocks of ON cells in n-th row of triangle generated by Wolfram's "Rule 30".

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 4, 5, 6, 6, 7, 7, 8, 7, 7, 8, 10, 9, 9, 11, 14, 12, 11, 12, 15, 16, 14, 15, 17, 14, 15, 17, 20, 18, 18, 18, 21, 21, 17, 19, 21, 20, 23, 22, 23, 22, 23, 21, 27, 30, 26, 27, 29, 29, 28, 28, 33, 31, 30, 31, 36, 32, 28, 29, 33, 33, 33, 35

Offset: 1

Alexandre Wajnberg, Sep 06 2005

Old name was "Number of trees appearing at n-th generation of a black cell following Wolfram's Rule 30 cellular automaton."

At each generation, "looking back", one can see "behind", groups (sort of black isles) of contiguous black cells which after a while appear to be trees growing. It should be possible to describe each one of them in terms of trees theory.

			a(1)=1 because one black cell;
a(2)=1 because there are now 3 contiguous black cell connected to the first one, which forms one only black surface;
a(3)=2 because two black cells are now connected to the preceding black surface and another black cell appears, which is isolated, so we have two separate black surfaces: 2.
From _Charlie Neder_, Feb 06 2019: (Start)
Rule 30 triangle begins:
                     1
                    111
                   11  1
                  11 1111
                 11  1   1
                11 1111 111
               11  1    1  1
              11 1111  111111
             11  1   111     1
and the number of blocks of ON cells in each row is 1, 1, 2, 2, 3, 3, 4, 3, 4, ... (End)

Charlie Neder, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rule 30.

Cf. A070950, A051023, A092539, A092540, A070952, A100053, A100054, A100055, A094603, A094604, A000225, A074890.

New name and a(17)-a(70) from Charlie Neder, Feb 06 2019

cp's OEIS Frontend

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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A094605 a(n) is the period of the n-th diagonal, from the right, of Rule 30 (begun from an initial black cell).

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A094603 a(n) is the length of the maximal sequence of rightmost black cells in the n-th row of Rule 30 (begun from an initial black cell).

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A110267 Total number of black cells at the first n generations of a single black cell following Wolfram's Rule 30 cellular automaton.

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A094606 Base-2 logarithm of A094605.

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A110266 Number of blocks of ON cells in n-th row of triangle generated by Wolfram's "Rule 30".

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