A267357 -id:A267357 - 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).

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)).

A328111 a(n) = A080069(n) OR A267357(n).

Original entry on oeis.org

1, 3, 15, 47, 191, 743, 2935, 12015, 47615, 190363, 737255, 3092431, 11777535, 48562151, 194672615, 778681963, 3117668351, 12677730147, 49850341191, 192901085003, 795560607711, 3243899871031, 12977889600367, 51055599708139, 204124618746111, 791262494980483, 3318011560984519, 12661179187462123, 52138250822737375, 212591566440951715, 836346216751952367, 3236342451194541807

Offset: 0

Antti Karttunen, Oct 05 2019

nonn

The pattern has a remarkably nice texture. A269174 gives the trajectory of 1-D Cellular Automaton rule 124 (which is a mirror image of rule 110), when started from a single alive cell. Trails of its evolution can be dimly discerned on the right hand side of given illustrations, while the left hand side shows the evolution of (left hand side of) iterated Dyck-path system A080069 unblemished.

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

Cf. A003986, A267357, A269174.
Cf. A080069, A080070, and also A079438 and A123050.
Cf. also A328103.

a(n) = A080069(n) OR A267357(n), where OR is bitwise-OR, A003986.

A269174 Formula for Wolfram's Rule 124 cellular automaton: a(n) = (n OR 2n) AND ((n XOR 2n) OR (n XOR 4n)).

Original entry on oeis.org

0, 3, 6, 7, 12, 15, 14, 11, 24, 27, 30, 31, 28, 31, 22, 19, 48, 51, 54, 55, 60, 63, 62, 59, 56, 59, 62, 63, 44, 47, 38, 35, 96, 99, 102, 103, 108, 111, 110, 107, 120, 123, 126, 127, 124, 127, 118, 115, 112, 115, 118, 119, 124, 127, 126, 123, 88, 91, 94, 95, 76, 79, 70, 67, 192, 195, 198, 199, 204, 207, 206, 203, 216

Offset: 0

Antti Karttunen, Feb 22 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32767
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A003986, A003987, A004198, A048724, A048725, A057889, A269173, A161903.
Cf. A269175.
Cf. A269176 (numbers not present in this sequence).
Cf. A269177 (same sequence sorted into ascending order, duplicates removed).
Cf. A269178 (numbers that occur only once).
Cf. A267357 (iterates from 1 onward).

package main
import "fmt"
func main() {
    for n:=0; n<=100; n++{
    fmt.Println((n|2*n)&((n^(2*n))|(n^(4*n))))}
} // Indranil Ghosh, Apr 19 2017

a[n_] := BitAnd[BitOr[n, 2n], BitOr[BitXor[n, 2n], BitXor[n, 4n]]];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 23 2020 *)

def a(n): return (n|2*n)&((n^(2*n))|(n^(4*n))) # Indranil Ghosh, Apr 19 2017

(define (A269174 n) (A004198bi (A163617 n) (A003986bi (A048724 n) (A048725 n))))

a(n) = A163617(n) AND A269173(n).
a(n) = A163617(n) AND (A048724(n) OR A048725(n)).
a(n) = (n OR 2n) AND ((n XOR 2n) OR (n XOR 4n)).
Other identities. For all n >= 0:
a(2*n) = 2*a(n).
a(n) = A057889(A161903(A057889(n))). [Rule 124 is the mirror image of rule 110.]
G.f.: (-3*x^3 - 2*x^2 - 3*x)/(x^4 - 1) + Sum_{k>=1}((2^(k + 1)*x^(2^k) - 2^(k + 1)*x^(14*2^(k - 2)))/((x^(2^(k + 2)) - 1)*(x - 1))). - Miles Wilson, Jan 25 2025

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

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A328111 a(n) = A080069(n) OR A267357(n).

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A269174 Formula for Wolfram's Rule 124 cellular automaton: a(n) = (n OR 2n) AND ((n XOR 2n) OR (n XOR 4n)).

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