A070952 -id:A070952 - cp's OEIS frontend

A246024 A070952(n)-n.

1, 2, 1, 3, 0, 4, -1, 5, -1, 3, 1, 3, 0, 6, -1, 7, -1, 2, 2, 5, 1, 2, 1, 5, 2, 2, 0, 6, 2, 5, 1, 8, -6, 6, -5, 11, -4, 7, 0, 6, 7, 0, 3, 6, -6, 10, -4, 4, -4, 4, -7, 8, 0, 7, -5, 10, 1, 3, -2, 10, 1, 9, -3, 15, 0, -9, -1, 2, 1, -2, 7, 7, 1, -5, 9, 3, 2, 10, 2, 7, 8, -2, -1, 11, -3, 16, -8, 1, 3

Offset: 0

N. J. A. Sloane, Aug 14 2014

sign

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Cf. A070952, A246027.

A246023 a(n) = A070952(2^n).

Original entry on oeis.org

3, 3, 4, 7, 15, 26, 64, 132, 253, 498, 1033, 2027, 4089, 8141, 16416, 32715, 65722, 131374, 262159, 524211, 1048630, 2097291, 4195039, 8387885

Offset: 0

N. J. A. Sloane, Aug 14 2014

Arises from studying behavior of 1-D CA defined by Rule 30 when started with a single ON cell.
I would very much like to see a recurrence or formula.
Subtracting 2^n gives 2, 1, 0, -1, -1, -6, 0, 4, -3, -14, 9, -21, -7, -51, ...

Cf. A070952, A246024, A246597.

a(14)-a(23) from Hiroaki Yamanouchi, Sep 12 2014

A151929 First differences of A070952.

Original entry on oeis.org

1, 2, 0, 3, -2, 5, -4, 7, -5, 5, -1, 3, -2, 7, -6, 9, -7, 4, 1, 4, -3, 2, 0, 5, -2, 1, -1, 7, -3, 4, -3, 8, -13, 13, -10, 17, -14, 12, -6, 7, 2, -6, 4, 4, -11, 17, -13, 9, -7, 9, -10, 16, -7, 8, -11, 16, -8, 3, -4, 13, -8, 9, -11, 19, -14, -8, 9, 4, 0, -2, 10, 1, -5, -5, 15, -5, 0, 9, -7, 6, 2, -9, 2, 13, -13, 20, -23, 10, 3, -5, 0, 27, -34, 22

Offset: 0

N. J. A. Sloane, Aug 10 2009

sign

Net increase in number of ON cells at generation n of 1-D CA using Rule 30.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (Added Aug 15 2009)
Index entries for sequences related to cellular automata

steps = 100; Join[{1}, Total /@ CellularAutomaton[30, {{1}, 0}, steps] // Differences] (* Jean-François Alcover, Oct 07 2013 *)

A246597 A070952(2^n-1).

Original entry on oeis.org

1, 3, 6, 12, 22, 39, 78, 132, 269, 526, 1036, 2067, 4162, 8252, 16387, 32662, 65372, 130953, 262212, 524550, 1047850, 2097353, 4192505, 8384471

Offset: 0

N. J. A. Sloane, Sep 06 2014

Arises from studying behavior of 1-D CA defined by Rule 30 when started with a single ON cell.
I would very much like to see a recurrence or formula.
Subtracting 2^n gives 0, 1, 2, 4, 6, 7, 14, 4, 13, 14, 12, 19, 66, 60, ...

Cf. A070952, A246024, A246023.

a(14)-a(23) from Hiroaki Yamanouchi, Sep 12 2014

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

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

Original entry on oeis.org

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

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 *)

A071049 Number of 1's in n-th generation of 1-D CA using Rule 110, started with a single 1.

Original entry on oeis.org

1, 2, 3, 3, 5, 3, 5, 6, 8, 5, 6, 8, 8, 8, 11, 11, 13, 9, 11, 11, 13, 14, 16, 14, 14, 13, 13, 17, 22, 20, 16, 17, 24, 19, 14, 19, 25, 18, 20, 25, 24, 19, 24, 31, 27, 26, 24, 22, 32, 31, 28, 24, 29, 34, 30, 31, 37, 34, 34, 36, 35, 34, 35, 36, 43, 40, 36, 38, 37, 39, 40

Offset: 0

Hans Havermann, May 26 2002

nonn

Number of 1's in n-th row of triangle in A070887.

Although the initial behavior is chaotic, it is an astonishing fact, pointed out by Wolfram [2002, p. 39], that after about three thousand terms all the irregularities disappear. - N. J. A. Sloane, May 15 2015

Matthew Cook, A Concrete View of Rule 110 Computation, in "The Complexity of Simple Programs", T. Neary, D. Woods, A. K. Seda, and N. Murphy (Eds.), 2008, pp. 31-55.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Vincenzo Librandi and N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (First 1000 terms from Vincenzo Librandi)
Matthew Cook, A Concrete View of Rule 110 Computation, arXiv:0906.3248 [cs.CC], 2009.
Matthew Cook, Universality in Elementary Cellular Automata, Complex Systems 15 (2004), 1-40.
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 110
Wikipedia, Rule 110
Index entries for sequences related to cellular automata

Cf. A070950, A070952, A070997, A071029, A071044, A151931, A246027.

A071049 := proc(n)
    add( A070887(n+1,k),k=1..n+1) ;
end proc:
seq(A071049(n),n=0..20) ; # R. J. Mathar, Feb 18 2015

Total /@ CellularAutomaton[110,{{1},0},100] (* N. J. A. Sloane, Aug 10 2009 *)

For n >= 2854, a(n+469) = -a(n+453) + a(n+256) + a(n+240) + a(n+229) + a(n+213) - a(n+16) - a(n). - N. J. A. Sloane, May 15 2015

Added references and links. - N. J. A. Sloane, Aug 09 2014

Changed offset to make consistent with A070952, etc. - N. J. A. Sloane, Aug 15 2014

A328106 Binary weight of A327971: a(n) = A000120(A110240(n) XOR A030101(A110240(n))).

Original entry on oeis.org

0, 0, 2, 2, 2, 4, 6, 4, 8, 10, 10, 8, 12, 8, 18, 6, 12, 26, 16, 18, 14, 18, 20, 22, 22, 26, 26, 38, 30, 26, 36, 26, 28, 36, 28, 18, 28, 42, 36, 32, 34, 40, 44, 38, 40, 50, 48, 48, 50, 58, 46, 56, 48, 42, 54, 48, 56, 56, 46, 54, 48, 52, 60, 58, 78, 74, 64, 60, 66, 74, 74, 64, 80, 74, 80, 62, 92, 62, 80, 70, 68, 100, 90, 82, 80, 92

Offset: 0

Antti Karttunen, Oct 05 2019

nonn

a(n) is the number of times the k-th cell from the left is different from the k-th cell from the right, at the generation n of Rule 30 1-D cellular automaton, when it is started from a single alive cell.

All terms are even.

			The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:
---------------------------------------------- a(n)
   0:              (1)                          0
   1:             1(1)1                         0
   2:            11(0)01                        2
   3:           110(1)111                       2
   4:          1100(1)0001                      2
   5:         11011(1)10111                     4
   6:        110010(0)001001                    6
   7:       1101111(0)0111111                   4
   8:      11001000(1)11000001                  8
   9:     110111101(1)001000111                10
  10:    1100100001(0)1111011001               10
  11:   11011110011(0)10000101111               8
  12:  110010001110(0)110011010001             12
  13: 1101111011001(1)1011100110111             8
When we count the times the k-th cell from the left is different from the k-th cell from the right, we obtain a(n). Note that the central cells (indicated with parentheses) do not affect the count, as the central cell is always equal to itself.

Cf. A000120, A070950, A110240, A144078, A269160, A280508, A327971.

Cf. also A070952, A328105, A328107, A328108, A328109.

A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A328106(n) = hammingweight(bitxor(A110240(n), A030101(A110240(n))));
\\ Use this one for writing b-files:
A328106write(up_to) = { my(s=1, n=0); for(n=0,up_to, write("b328106.txt", n, " ", hammingweight(bitxor(s, A030101(s)))); s = A269160(s)); };

a(n) = A000120(A327971(n)) = A144078(A110240(n)) = A000120(A280508(A110240(n))).

a(n) = Sum_{i=0..2n} abs(A070950(n,i)-A070950(n,n-i)).

A070951 Number of 0's in n-th row of triangle in A070950.

Original entry on oeis.org

0, 0, 2, 1, 5, 2, 8, 3, 10, 7, 10, 9, 13, 8, 16, 9, 18, 16, 17, 15, 20, 20, 22, 19, 23, 24, 27, 22, 27, 25, 30, 24, 39, 28, 40, 25, 41, 31, 39, 34, 34, 42, 40, 38, 51, 36, 51, 44, 53, 46, 58, 44, 53, 47, 60, 46, 56, 55, 61, 50, 60, 53, 66, 49, 65, 75, 68, 66, 68, 72

Offset: 0

N. J. A. Sloane, May 19 2002

a(n) + A070952(n) = 2*n + 1. - Reinhard Zumkeller, Jun 07 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A070950, A070952.

a070951 = length . filter (== 0) . a070950_row
-- Reinhard Zumkeller, Jun 06 2013

steps = 100; Count[#, 0]& /@ MapIndexed[ Take[#1, {steps - First[#2] + 2, steps + First[#2]}]&, CellularAutomaton[30, {{1}, 0}, steps]] (* Jean-François Alcover, Oct 07 2013 *)

More terms from Hans Havermann, May 26 2002

cp's OEIS Frontend

A246024 A070952(n)-n.

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A246023 a(n) = A070952(2^n).

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A151929 First differences of A070952.

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A246597 A070952(2^n-1).

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

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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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A071049 Number of 1's in n-th generation of 1-D CA using Rule 110, started with a single 1.

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