A328107 -id:A328107 - cp's OEIS frontend

A327973 Bitwise XOR of two successive generations (centrally aligned) in the trajectory of rule 30 started from a lone 1 cell: a(n) = A110240(n) XOR 2*A110240(n-1).

5, 23, 93, 335, 1493, 5351, 23853, 85951, 382405, 1369943, 6103965, 21996687, 97906325, 350709671, 1562619373, 5631262591, 25064000389, 89782414999, 400033474525, 1441615751887, 6416397448021, 22984338788455, 102408232210605, 369052763468095, 1642598765228869, 5883986891577303, 26216498605021469, 94477513773305103

Offset: 1

Antti Karttunen, Oct 03 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Antti Karttunen, Terms a(1)-a(256) drawn as binary strings, with 1 bit = 3x3 pixels resolution
Antti Karttunen, Terms a(1)-a(1024) drawn as binary strings, with 1 bit = 1 pixel resolution
Index entries for sequences related to cellular automata

Cf. A110240, A269160, A327974 (gives the middle bit), A328107 (binary weight of terms).

Cf. also A327971, A327972, 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)));
A327973(n) = bitxor(A110240(n), 2*A110240(n-1));
\\ Use this one for writing b-files:
A327973write(up_to) = { my(s=1, t, n=0); for(n=1,up_to, t = A269160(s); write("b327973.txt", n, " ", bitxor(2*s, t)); s = t); };

def A269160(n): return(n^((n<<1)|(n<<2)))
def genA327973():
    '''Yield successive terms of A327973.'''
    s = 1
    while True:
       t = A269160(s)
       yield (t^(s<<1))
       s = t

a(n) = A110240(n) XOR 2*A110240(n-1).

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).

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

A327973 Bitwise XOR of two successive generations (centrally aligned) in the trajectory of rule 30 started from a lone 1 cell: a(n) = A110240(n) XOR 2*A110240(n-1).

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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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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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