A328109 -id:A328109 - cp's OEIS frontend

A328103 Bitwise XOR of trajectories of rule 30 and rule 124, when both are started from a lone 1 cell: a(n) = A110240(n) XOR A267357(n).

0, 4, 30, 100, 398, 1748, 6510, 28628, 102590, 456132, 1642078, 7289764, 26336590, 116802708, 420215854, 1865678868, 6741198206, 29904470916, 107568473246, 477629808612, 1725756768270, 7655529847380, 27537572248046, 122273029571156, 441793665700414, 1959816793456452, 7049616389341662, 31301899019407908, 113099196716630990, 501713069953322004

Offset: 0

Antti Karttunen, Oct 05 2019

nonn

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 cellular automata

Cf. A003987, A110240, A267357, A269160, A269174, A328109 (binary weight of terms).

Cf. also A327971, A327972, A327973, A327976, A328104 for other such combinations, and also A328111.

A269160(n) = bitxor(n, bitor(2*n, 4*n));
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)));
A328103(n) = bitxor(A110240(n),A267357(n));
\\ Use this one for writing b-files:
A328103write(up_to) = { my(s1=1, s2=1); for(n=0,up_to, write("b328103.txt", n, " ", bitxor(s1, s2)); s1 = A269160(s1); s2 = A269174(s2)); };

def A269160(n): return(n^((n<<1)|(n<<2)))
def A269174(n): return((n|(n<<1))&((n^(n<<1))|(n^(n<<2))))
def genA328103():
    '''Yield successive terms of A328103.'''
    s1 = 1
    s2 = 1
    while True:
       yield (s1^s2)
       s1 = A269174(s1)
       s2 = A269160(s2)

a(n) = A110240(n) XOR A267357(n), where XOR is bitwise exclusive or (A003987).

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)).

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))).

cp's OEIS Frontend

A328103 Bitwise XOR of trajectories of rule 30 and rule 124, when both are started from a lone 1 cell: a(n) = A110240(n) XOR A267357(n).

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