A328106 -id:A328106 - 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

A327971 Bitwise XOR of trajectories of rule 30 and its mirror image, rule 86, when both are started from a lone 1 cell: a(n) = A110240(n) XOR A265281(n).

Original entry on oeis.org

0, 0, 10, 20, 130, 396, 2842, 4420, 38610, 124220, 684490, 1385044, 8891330, 26281036, 192525274, 269101060, 2454365330, 8588410876, 43860512138, 89059958420, 551714970626, 1663794165260, 12235920695450, 19683098342340, 164315052318034, 538162708968636, 2894532467106378, 6192136868790228, 37503903254935874, 114926395086966988, 814341599153559130

Offset: 0

Antti Karttunen, Oct 03 2019

nonn

Each term is a binary palindrome when its trailing zeros (in base 2) are omitted, that is, a term of A057890.

Compare the binary string illustrations drawn for the first 1024 terms of this sequence and for A327976, which has almost the same definition.

Antti Karttunen, Table of n, a(n) for n = 0..1023
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
Index entries for sequences related to binary expansion of n
Index entries for sequences related to cellular automata

Cf. A003987, A030101, A057890, A110240, A265281, A280508, A328106 (binary weight of terms).

Cf. also A327972, A327973, 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)));
A269161(n) = bitxor(4*n, bitor(2*n, n));
A265281(n) = if(!n,1,A269161(A265281(n-1)));
A327971(n) = bitxor(A110240(n), A265281(n));

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A327971write(up_to) = { my(s=1, n=0); for(n=0,up_to, write("b327971.txt", n, " ", bitxor(s, A030101(s))); s = A269160(s)); };

def A269160(n): return(n^((n<<1)|(n<<2)))
def A269161(n): return((n<<2)^((n<<1)|n))
def genA327971():
    '''Yield successive terms of A327971.'''
    s1 = 1
    s2 = 1
    while True:
       yield (s1^s2)
       s1 = A269160(s1)
       s2 = A269161(s2)

a(n) = A110240(n) XOR A265281(n).
a(n) = A280508(A110240(n)) = A110240(n) XOR A030101(A110240(n)).
a(n) = A280508(A265281(n)) = A265281(n) XOR A030101(A265281(n)).
For n >= 1, a(n) = (1/2) * (A327973(n-1) XOR A327976(n-1)).

A328105 Binary weight of A328104: a(n) = A000120(A110240(n) OR 2*A110240(n)).

Original entry on oeis.org

2, 4, 5, 8, 7, 12, 9, 15, 11, 17, 17, 20, 19, 26, 21, 29, 22, 27, 30, 33, 30, 34, 37, 40, 37, 39, 41, 49, 44, 49, 48, 53, 41, 56, 49, 64, 50, 62, 59, 66, 64, 60, 66, 69, 61, 77, 65, 73, 67, 74, 70, 89, 78, 87, 78, 94, 85, 88, 89, 100, 91, 101, 95, 110, 92, 85, 98, 102, 102, 102, 115, 109, 101, 105, 121, 118, 121, 129

Offset: 0

Antti Karttunen, Oct 05 2019

nonn

Cf. A000120, A110240, A163617, A269160, A328104.

Cf. also A070952, A328106, A328107, A328108, A328109.

A269160(n) = bitxor(n, bitor(2*n, 4*n));
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A163617(n) = bitor(n,n<<1);
A328105(n) = hammingweight(A163617(A110240(n)));
\\ Use this one for writing b-files:
A328105write(up_to) = { my(s=1, n=0); for(n=0,up_to, write("b328105.txt", n, " ", hammingweight(bitor(s, s<<1))); s = A269160(s)); };

a(n) = A000120(A328104(n)) = A000120(A163617(A110240(n))).

For all n >= 0, A070952(a) < a(n) <= 2*A070952(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)).

A328108 Binary weight of A327976.

Original entry on oeis.org

2, 4, 3, 6, 5, 12, 7, 11, 9, 13, 7, 12, 13, 20, 15, 23, 16, 19, 22, 25, 20, 28, 19, 30, 29, 39, 27, 29, 32, 37, 32, 37, 29, 38, 37, 38, 36, 44, 47, 44, 42, 46, 42, 53, 41, 49, 53, 47, 45, 58, 52, 55, 56, 65, 66, 60, 67, 56, 61, 64, 63, 77, 59, 66, 60, 67, 72, 72, 64, 84, 57, 81, 63, 79, 67, 92, 77, 77, 74, 80, 81, 88, 77

Offset: 1

Antti Karttunen, Oct 05 2019

nonn

Cf. A000120, A030101, A110240, A265281, A269160, A327976.

Cf. also A070952, A328105, A328106, A328107, 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));
A328108(n) = hammingweight(bitxor(A110240(n), 2*A030101(A110240(n-1))));
\\ Use this one for writing b-files:
A328108write(up_to) = { my(s=1, t, n=0); for(n=1,up_to, t = A269160(s); write("b328108.txt", n, " ", hammingweight(bitxor(2*A030101(s), t))); s = t); };

a(n) = A000120(A327976(n)).
a(n) = A000120(A110240(n) XOR 2*A265281(n-1)).
a(n) = A000120(A110240(n) XOR 2*A030101(A110240(n-1))).

A328109 Binary weight of A328103: a(n) = A000120(A110240(n) XOR A267357(n)).

Original entry on oeis.org

0, 1, 4, 3, 5, 6, 8, 10, 9, 11, 11, 14, 14, 13, 16, 11, 18, 16, 17, 25, 18, 21, 25, 24, 22, 30, 25, 28, 30, 26, 33, 34, 36, 34, 33, 37, 37, 44, 38, 44, 51, 38, 43, 48, 45, 57, 38, 47, 50, 52, 49, 61, 53, 56, 63, 58, 56, 54, 60, 59, 64, 54, 60, 66, 69, 60, 67, 69, 72, 68, 75, 74, 77, 68, 78, 76, 75, 78, 72, 81, 80, 91, 78

Offset: 0

Antti Karttunen, Oct 05 2019

nonn

Cf. A000120, A110240, A267357, A328103.

Cf. also A328105, A328106, A328107, A328108.

A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A269174(n) = bitand(bitor(n,n<<1),bitor(bitxor(n,n<<1),bitxor(n,n<<2)));
A267357(n) = if(!n,1,A269174(A267357(n-1)));
A328109(n) = hammingweight(bitxor(A110240(n),A267357(n)));
\\ Use this one for writing b-files:
A328109write(up_to) = { my(s1=1, s2=1); for(n=0,up_to, write("b328109.txt", n, " ", hammingweight(bitxor(s1, s2))); s1 = A269160(s1); s2 = A269174(s2)); };

a(n) = A000120(A328103(n)) = A000120(A110240(n) XOR A267357(n)).

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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A327971 Bitwise XOR of trajectories of rule 30 and its mirror image, rule 86, when both are started from a lone 1 cell: a(n) = A110240(n) XOR A265281(n).

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A328105 Binary weight of A328104: a(n) = A000120(A110240(n) OR 2*A110240(n)).

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A328107 Binary weight of A327973.

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A328108 Binary weight of A327976.

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A328109 Binary weight of A328103: a(n) = A000120(A110240(n) XOR A267357(n)).

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