A100053 -id:A100053 - cp's OEIS frontend

A100054 Records in A100053.

0, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 23, 24, 26, 28, 33, 35

Offset: 0

Eric W. Weisstein, Oct 31 2004

Eric Weisstein's World of Mathematics, Rule 30
Index entries for sequences related to cellular automata

clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; t = Max /@ Map[Length@ # &, Map[Split@ # &, Map[clip, CellularAutomaton[30, {{1}, 0}, 5000]]] /. 1 -> Nothing, {2}]; a = {0}; Do[If[t[[n]] > Max@ a, AppendTo[a, t[[n]]]], {n, Length@ t}]; a (* Michael De Vlieger, Oct 06 2015 *)

A100055 Where records occur in A100053.

Original entry on oeis.org

0, 2, 4, 6, 8, 16, 32, 43, 46, 64, 128, 256, 512, 1024, 2048, 4096, 8192

Offset: 0

Eric W. Weisstein, Nov 01 2004

Eric Weisstein's World of Mathematics, Rule 30

Cf. A100053, A100054.

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

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

A317530 a(n) is that generation of the rule-30 1D cellular automaton started from a single ON cell in which n successive OFF cells appears for the first time.

Original entry on oeis.org

4, 3, 5, 7, 9, 17, 45, 33, 44, 67, 47, 66, 130, 65, 129, 517, 260, 516, 259, 515, 258, 514, 257, 513, 4101, 1025, 4100, 2049, 4099, 16388, 4098, 16387, 4097, 16386, 8193, 16385, 262150, 32769, 262149, 65537, 262148, 131073, 262147, 524291, 262146, 524290, 262145, 524289

Offset: 1

Philipp O. Tsvetkov, Jul 30 2018

nonn

OFF cells outside the triangle of active cells are ignored.

This is also the position at which n appears for the first time in A100053.

			The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle:
1                          .
2                        . . .
3                      . . 0 0 .
4                    . . 0 . . . .
5                  . . 0 0 . 0 0 0 .
6                . . 0 . . . . 0 . . .
7              . . 0 0 . 0 0 0 0 . 0 0 .
8            . . 0 . . . . 0 0 . . . . . .
9          . . 0 0 . 0 0 0 . . . 0 0 0 0 0 .
10       . . 0 . . . . 0 . . 0 0 . 0 0 0 . . .
11     . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 .
12   . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . .
13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 .
1 OFF cell (0) appears for the first time in generation (line) 4, thus a(1) = 4;
2 consecutive OFF cells (00) appear for the first time in generation (line) 3, thus a(2) = 3;
5 consecutive OFF cells (00000) appear for the first time in generation (line) 9, thus a(5) = 9;
...

Rémy Sigrist, C program for A317530

Cf. A100053.

C
```
See Links section.
```

CellularAutomaton[30, {{1}, 0}, 10000];
(Reverse[Internal`DeleteTrailingZeros[
      Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;
Table[Position[(Max@FoldList[If[#2 == 0, #1 + 1, 0] &, 0, #]) & /@ %, i] //
   Flatten // First, {i, 1, 29}]

a(30)-a(36) from Robert G. Wilson v, Aug 07 2018

Data corrected and more terms from Rémy Sigrist, Jul 06 2020

A319610 a(n) is the minimal number of successive OFF cells that appears in n-th generation of rule-30 1D cellular automaton started from a single ON cell.

Original entry on oeis.org

0, 0, 2, 1, 2, 1, 2, 1, 2, 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, 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

Offset: 0

Philipp O. Tsvetkov, Sep 24 2018

nonn

OFF cells outside the triangle of active cells are ignored.

			The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle:
1                          .                             a(1)= (0)
2                        . . .                           a(2)= (0)
3                      . . 0 0 .                         a(3)= (2)
4                    . . 0 . . . .                       a(4)= (1)
5                  . . 0 0 . 0 0 0 .                     a(5)= (2)
6                . . 0 . . . . 0 . . .                   a(6)= (1)
7              . . 0 0 . 0 0 0 0 . 0 0 .                 a(7)= (2)
8            . . 0 . . . . 0 0 . . . . . .               a(8)= (1)
9          . . 0 0 . 0 0 0 . . . 0 0 0 0 0 .             a(9)= (2)
10       . . 0 . . . . 0 . . 0 0 . 0 0 0 . . .           a(10)=(1)
11     . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 .         a(11)=(1)
12   . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . .       a(12)=(1)
13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 .     a(13)=(1)

Charlie Neder, Repeating pattern of length-1 runs
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A100053.

CellularAutomaton[30, {{1}, 0}, 200];
(Reverse[Internal`DeleteTrailingZeros[Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;
Table[Length /@ Select[%[[i]] // Split, Total[#] == 0 &] // Min, {i, 1, % // Length}]

G.f.: x (x + x/(1 - x) + x^3 + x^5 + x^7) (conjectured).
For n > 9, a(n)=1 at least up to n = 20000.
It is conjectured that for all n>=10, a(n)=1.
A period-4 pattern of length-1 runs starting at row 26 forces a(n) = 1 for all n >= 26 (see image). - Charlie Neder, Dec 15 2018

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

A319658 a(n) is the minimal number of successive ON cells that appears in n-th generation of rule-30 1D cellular automaton started from a single ON cell.

Original entry on oeis.org

1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 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

Offset: 1

Philipp O. Tsvetkov, Sep 25 2018

nonn

			The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle:
1                          .                             a(1)= (1)
2                        . . .                           a(2)= (3)
3                      . . 0 0 .                         a(3)= (1)
4                    . . 0 . . . .                       a(4)= (2)
5                  . . 0 0 . 0 0 0 .                     a(5)= (1)
6                . . 0 . . . . 0 . . .                   a(6)= (2)
7              . . 0 0 . 0 0 0 0 . 0 0 .                 a(7)= (1)
8            . . 0 . . . . 0 0 . . . . . .               a(8)= (2)
9          . . 0 0 . 0 0 0 . . . 0 0 0 0 0 .             a(9)= (1)
10       . . 0 . . . . 0 . . 0 0 . 0 0 0 . . .           a(10)=(1)
11     . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 .         a(11)=(1)
12   . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . .       a(12)=(1)
13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 .     a(13)=(1)

Charlie Neder, Repeating pattern of length-1 runs
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A319610, A319610, A100053.

CellularAutomaton[30, {{1}, 0}, 100];
(Reverse[Internal`DeleteTrailingZeros[
      Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;
Table[Length /@ Select[%[[i]] // Split, Total[#] > 0 &] // Min, {i,
  1, % // Length}]

G.f.: 1/(1 - x) + 2 x + x^3 + x^5 + x^7 + x^13 (conjectured).
For n > 14, a(n)=1 at least until n = 10000.
It is conjectured that for all n >= 15, a(n)=1.
A period-4 pattern of length-1 runs beginning on row 19 forces a(n) = 1 for all n >= 19 (see image). - Charlie Neder, Dec 15 2018

A319611 a(n) is the number of gaps in the n-th generation of the rule-30 1D cellular automaton started from a single ON.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 2, 3, 4, 5, 5, 6, 6, 7, 6, 6, 7, 9, 8, 8, 10, 13, 11, 10, 11, 14, 15, 13, 14, 16, 13, 14, 16, 19, 17, 17, 17, 20, 20, 16, 18, 20, 19, 22, 21, 22, 21, 22, 20, 26, 29, 25, 26, 28, 28, 27, 27, 32, 30, 29, 30, 35, 31, 27, 28, 32, 32, 32, 34, 37, 30, 27, 36, 37, 39, 42, 41, 43, 41, 34

Offset: 0

Philipp O. Tsvetkov, Sep 24 2018

nonn

OFF cells outside the triangle of active cells are ignored.

			The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle:
1                          .                             a(1)= (0)
2                        . . .                           a(2)= (0)
3                      . . 0 0 .                         a(3)= (1)
4                    . . 0 . . . .                       a(4)= (1)
5                  . . 0 0 . 0 0 0 .                     a(5)= (2)
6                . . 0 . . . . 0 . . .                   a(6)= (2)
7              . . 0 0 . 0 0 0 0 . 0 0 .                 a(7)= (3)
8            . . 0 . . . . 0 0 . . . . . .               a(8)= (2)
9          . . 0 0 . 0 0 0 . . . 0 0 0 0 0 .             a(9)= (3)
10       . . 0 . . . . 0 . . 0 0 . 0 0 0 . . .           a(10)=(4)
11     . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 .         a(11)=(5)
12   . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . .       a(12)=(5)
13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 .     a(13)=(6)

Cf. A100053.

CellularAutomaton[30, {{1}, 0}, 200];
(Reverse[Internal`DeleteTrailingZeros[
      Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;
Table[
Length /@ Select[%[[i]] // Split, Total[#] == 0 &] // Length,
{i, 1, % // Length}
]

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A100054 Records in A100053.

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A100055 Where records occur in A100053.

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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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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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A317530 a(n) is that generation of the rule-30 1D cellular automaton started from a single ON cell in which n successive OFF cells appears for the first time.

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A319610 a(n) is the minimal number of successive OFF cells that appears in n-th generation of rule-30 1D cellular automaton started from a single ON cell.

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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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A319658 a(n) is the minimal number of successive ON cells that appears in n-th generation of rule-30 1D cellular automaton started from a single ON cell.

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