A110240 -id:A110240 - 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).

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).

Original entry on oeis.org

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).

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)).

A327976 Bitwise XOR of trajectories (centrally aligned) of rule 30, and its mirror image, rule 86, when both are started from a lone 1-bit, with the latter delayed by one step: a(n) = A110240(n) XOR 2*A265281(n-1).

Original entry on oeis.org

5, 23, 73, 359, 1233, 6143, 19225, 93495, 325729, 1518895, 4833289, 23453735, 81443089, 398815039, 1271974489, 6168932215, 21231239841, 99197620591, 314863189193, 1541326542823, 5312985402193, 26258203294847, 82884499362201, 400683454289591, 1406328980294113, 6532877164215983, 20744329255918985, 100303645024039591

Offset: 1

Antti Karttunen, Oct 04 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, A265281, A269160, A269161, A030101, A327974 (gives the middle bit), A328108 (binary weight).

Cf. also A327971, A327972, A327973, 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));
A269161(n) = bitxor(4*n, bitor(2*n, n));
A265281(n) = if(!n,1,A269161(A265281(n-1)));
A327976(n) = bitxor(A110240(n), 2*A265281(n-1));
\\ Use this one for writing b-files:
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A327976write(up_to) = { my(s=1, t, n=0); for(n=1,up_to, t = A269160(s); write("b327976.txt", n, " ", bitxor(2*A030101(s), t)); s = t); };

def A269160(n): return(n^((n<<1)|(n<<2)))
def A269161(n): return((n<<2)^((n<<1)|n))
def genA327976():
    '''Yield successive terms of A327976.'''
    s1 = 1
    s2 = 1
    while True:
       s1 = A269160(s1)
       yield (s1^(s2<<1))
       s2 = A269161(s2)

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

A328104 a(n) = A163617(A110240(n)) = A110240(n) OR 2*A110240(n).

Original entry on oeis.org

3, 15, 59, 255, 947, 4095, 15131, 65407, 242627, 1048271, 3874811, 16743551, 62119411, 268369791, 991927259, 4286447359, 15902689155, 68701773199, 253935222715, 1097330432511, 4071076396851, 17587676696575, 65007550988187, 280916526002175, 1042196361379523, 4502448248917967, 16641933085980923, 71914639532751871

Offset: 0

Antti Karttunen, Oct 04 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. A003986, A051023, A110240, A269160, A163617, A328105 (binary weight of terms).

Cf. also A327971, A327972, A327973, A327976, A328103 for other such combinations.

A269160(n) = bitxor(n, bitor(2*n, 4*n));
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A163617(n) = bitor(n,n<<1);
A328104(n) = A163617(A110240(n));
\\ Use this one for writing b-files:
A328104write(up_to) = { my(s=1, n=0); for(n=0,up_to, write("b328104.txt", n, " ", bitor(s, s<<1)); s = A269160(s)); };

def A269160(n): return(n^((n<<1)|(n<<2)))
def genA328104():
    '''Yield successive terms of A328104.'''
    s = 1
    while True:
       yield (s|(s<<1))
       s = A269160(s)

a(n) = A163617(A110240(n)) = A110240(n) OR 2*A110240(n).

a(n) = (1/2) * (A110240(n) XOR A110240(1+n)).

A327972 Bitwise XOR of trajectories of rule 30 and rule 150, when both are started from a lone 1 cell: a(n) = A110240(n) XOR A038184(n).

Original entry on oeis.org

0, 0, 12, 4, 128, 384, 3404, 740, 37056, 127296, 794316, 286532, 8510656, 25560896, 224057484, 42076324, 2446214016, 8430013568, 51732969356, 18062215300, 553213409792, 1655549411840, 14630859361996, 3227756349540, 159219183713088, 546944274202816, 3411332163636556, 1231354981057220, 36554500089286208, 109782277571646400, 962314238681316620

Offset: 0

Antti Karttunen, Oct 03 2019

nonn

Cf. A003987, A038184, A048727, A110240, A269160.

Cf. also A327971, A327973, A327976, A328103, A328104 for other such combinations.

A048727(n) = bitxor(n, bitxor(2*n, 4*n)); \\ From A048727
A038184(n) = if(!n,1,A048727(A038184(n-1)));
A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
A110240(n) = if(!n,1,A269160(A110240(n-1)));
A327972(n) = bitxor(A038184(n), A110240(n));
\\ Use this one for writing b-files:
A327972write(up_to) = { my(s1=1, s2=1); for(n=0,up_to, write("b327972.txt", n, " ", bitxor(s1, s2)); s1 = A048727(s1); s2 = A269160(s2)); };

def A048727(n): return(n^(n<<1)^(n<<2))
def A269160(n): return(n^((n<<1)|(n<<2)))
def genA327972():
    '''Yield successive terms of A327972.'''
    s1 = 1
    s2 = 1
    while True:
       yield (s1^s2)
       s1 = A269160(s1)
       s2 = A048727(s2)

a(n) = A038184(n) XOR A110240(n).

Conjecture: for n > 1, floor(log_2(a(n))) = 2*n - (1,2,1,4,1,2,1,5 according as n == 0..7 (mod 8), respectively). - Alan Michael Gómez Calderón, Mar 02 2023

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).

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)).

A269160 Formula for Wolfram's Rule 30 cellular automaton: a(n) = n XOR (2n OR 4n).

Original entry on oeis.org

0, 7, 14, 13, 28, 27, 26, 25, 56, 63, 54, 53, 52, 51, 50, 49, 112, 119, 126, 125, 108, 107, 106, 105, 104, 111, 102, 101, 100, 99, 98, 97, 224, 231, 238, 237, 252, 251, 250, 249, 216, 223, 214, 213, 212, 211, 210, 209, 208, 215, 222, 221, 204, 203, 202, 201, 200, 207, 198, 197, 196, 195, 194, 193, 448, 455, 462

Offset: 0

Antti Karttunen, Feb 20 2016

nonn

Take n, write it in binary, see what Rule 30 would do to that state, convert it to decimal: that is a(n). For example, we can see in A110240 that 7 = 111_2 becomes 25 = 11001_2 under Rule 30, which is shown here by a(7) = 25. - N. J. A. Sloane, Nov 25 2016
The sequence is injective: no value occurs more than once.
Fibbinary numbers (A003714) give all integers n>=0 for which a(n) = A048727(n) and for which a(n) = A269161(n).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Eric Weisstein's World of Mathematics, Rule 30
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A003714, A003986, A003987, A057889, A070939.
Cf. A110240 (iterates starting from 1).
Cf. A269162 (left inverse).
Cf. A269163 (same sequence sorted into ascending order).
Cf. A269164 (values missing from this sequence).
Cf. also A048727, A269161.

a[n_] := BitXor[n, BitOr[2n, 4n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 23 2016 *)

a(n) = bitxor(n, bitor(2*n, 4*n)); \\ Michel Marcus, Feb 23 2016

def A269160(n): return n^(n<<1 | n<<2) # Chai Wah Wu, Jun 29 2022

(define (A269160 n) (A003987bi n (A003986bi (* 4 n) (* 2 n)))) ;; Where A003986bi and A003987bi are implementation of dyadic functions giving bitwise-OR (A003986) and bitwise-XOR (A003987) of their arguments.

a(n) = n XOR (2n OR 4n) = A003987(n, A003986(2*n, 4*n)).
Other identities. For all n >= 0:
a(2*n) = 2*a(n).
a(n) = A057889(A269161(A057889(n))). [Rule 30 is the mirror image of rule 86.]
A269162(a(n)) = n.
For all n >= 1:
A070939(a(n)) - A070939(n) = 2. [The binary length of a(n) is two bits longer than that of n for all nonzero values.]
G.f.: (3*x + 2*x^2 +x^3)/(1 - x^4) + Sum_{k>=1}(2^(k + 1)*x^(2^(k - 1))/((1 + x^(2^(k + 1)))*(1 - x))). - Miles Wilson, Jan 24 2025

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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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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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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A327976 Bitwise XOR of trajectories (centrally aligned) of rule 30, and its mirror image, rule 86, when both are started from a lone 1-bit, with the latter delayed by one step: a(n) = A110240(n) XOR 2*A265281(n-1).

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A328104 a(n) = A163617(A110240(n)) = A110240(n) OR 2*A110240(n).

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A327972 Bitwise XOR of trajectories of rule 30 and rule 150, when both are started from a lone 1 cell: a(n) = A110240(n) XOR A038184(n).

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PARI

Python

Formula

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

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PARI

Formula

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

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