A269174 -id:A269174 - cp's OEIS frontend

A269175 a(n) = number of distinct k for which A269174(k) = n; number of finite predecessors for pattern encoded in the binary expansion of n in Wolfram's Rule 124 cellular automaton.

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 2, 1, 0, 2, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 0, 0, 2, 1, 1

Offset: 0

Antti Karttunen, Feb 22 2016

nonn

At positions A000225 seems to occur the record values of this sequence: 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, ... which seem to match with A000931 (Padovan sequence), or more exactly, with A182097 (Number of compositions (ordered partitions) into parts 2 and 3). Note that these values give also the number of predecessors for each "repunit-pattern" (2^n)-1 in Rule 110 cellular automaton, as rules 110 and 124 are mirror images of each other.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A000225, A000931, A182097, A269174.

Cf. A269176 (indices of zeros), A269177 (of nonzeros), A269178 (of ones).

(definec (A269175 n) (let loop ((p 0) (s 0)) (cond ((> p n) s) (else (loop (+ 1 p) (+ s (if (= n (A269174 p)) 1 0))))))) ;; Very straightforward and very slow.
;; Somewhat optimized version:
(definec (A269175 n) (if (zero? n) 1 (let ((nwid-1 (- (A000523 n) 1))) (let loop ((p (if (< n 2) 0 (A000079 nwid-1))) (s 0)) (cond ((> (A000523 p) nwid-1) s) (else (loop (+ 1 p) (+ s (if (= n (A269174 p)) 1 0)))))))))

A269176 Numbers not present in A269174; indices of zeros in A269175.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 21, 23, 25, 26, 29, 32, 33, 34, 36, 37, 39, 40, 41, 42, 43, 45, 46, 49, 50, 52, 53, 57, 58, 61, 64, 65, 66, 68, 69, 71, 72, 73, 74, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 92, 93, 97, 98, 100, 101, 104, 105, 106, 109, 113, 114, 116, 117, 121, 122, 125, 128, 129

Offset: 1

Antti Karttunen, Feb 22 2016

nonn

Numbers n for which there is no any k such that A269174(k) = n.

These are binary representations (shown in decimal) of Garden of Eden patterns in Wolfram's Rule 124 cellular automaton if infinite predecessors are forbidden.

Antti Karttunen, Table of n, a(n) for n = 1..11672
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A269174, A269175.

Cf. A269177 (complement).

A332465 Numbers n for which A269174(sigma(n)) is equal to 2*A269174(n).

Original entry on oeis.org

6, 28, 348, 496, 732, 886, 2924, 3573, 4972, 5448, 7544, 8128, 23388, 54842, 66928, 89200, 92296, 109786, 118064, 121552, 349512, 356488, 367472, 550432, 634784, 839984, 842452, 1234048, 1561408, 1797496, 2154584, 2364832, 2788808, 2927992, 3451456, 3585328, 5952364, 5991852, 6687136, 8238752, 10594336, 11210712, 11261020

Offset: 1

Antti Karttunen, Feb 16 2020

nonn

Numbers n such that A332464(n) is equal to A269174(2*n).
There are only three odd terms <= 2^32 among the first 113 terms of this sequence: 3573, 29255157, 936109557. Because A269174 preserves the 2-adic valuation of its argument, all such odd terms are of the form 4m+1, and must be present in A191218. Incidentally, these three terms are also present in A228058, but not in A332227.
See from the graph how unevenly the terms appear. Compare also the scatter plots of A269174 and A332464, also of a similar sequence A332445.

			          k   factorization        sigma(k)     A269174(sigma(k)) = A269174(2*k)
        348 = 2^2 * 3 * 29         840          2008,
       3573 = 3^2 * 397            5174         15486,
   29255157 = 3^2 * 3250573        42257462     126737534,
  936109557 = 3^2 * 104012173      1352158262   4055424126.

Cf. A000203, A269174, A332464.
Cf. A000396 (a subsequence).
Cf. also A191218, A228058, A332227, A332445, A332446.

b[n_] := BitAnd[BitOr[n, 2n], BitOr[BitXor[n, 2n], BitXor[n, 4n]]];
okQ[n_] := b[DivisorSigma[1, n]] == 2 b[n];
Reap[For[n = 1, n <= 12*10^6, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 23 2020 *)

A269174(n) = bitand(bitor(n,n<<1),bitor(bitxor(n,n<<1),bitxor(n,n<<2)));
isA332465(n) = (A269174(sigma(n))==2*A269174(n));

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).

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)).

A161903 Convert n into a sequence of binary digits, apply one step of the rule 110 cellular automaton, and interpret the results as a binary integer.

Original entry on oeis.org

0, 3, 6, 7, 12, 15, 14, 13, 24, 27, 30, 31, 28, 31, 26, 25, 48, 51, 54, 55, 60, 63, 62, 61, 56, 59, 62, 63, 52, 55, 50, 49, 96, 99, 102, 103, 108, 111, 110, 109, 120, 123, 126, 127, 124, 127, 122, 121, 112, 115, 118, 119, 124, 127, 126, 125, 104, 107, 110, 111, 100, 103, 98, 97, 192, 195, 198, 199, 204, 207, 206, 205, 216, 219, 222, 223, 220, 223, 218, 217, 240, 243, 246, 247, 252, 255, 254, 253, 248, 251, 254, 255, 244, 247, 242, 241, 224, 227, 230, 231, 236

Offset: 0

Ben Branman, Jan 30 2011

a(a(a(...1))) (n times) gives A006978(n)

			For n=19, the evolution after one step is
0, 1, 0, 0, 1, 1  (n=19)
1, 1, 0, 1, 1, 1  (a(n)=55)
So a(n)=55.

T. D. Noe, Table of n, a(n) for n = 0..1023
Index entries for sequences related to cellular automata
Eric Weisstein's World of Mathematics, Rule 110
Wikipedia, Rule 110

Cf. A006978, A070887, A071049, A180001, A186083, A204371.

Cf. also A269160, A269174, A269175.

a[n_] :=
FromDigits[
  Drop[Part[CellularAutomaton[110, {IntegerDigits[n, 2], 0}], 1], -1],
   2];Table[a[n],{n,0,100}]

a(n) = A057889(A269174(A057889(n))). - Antti Karttunen, Jun 02 2018

A269177 Numbers that have a finite predecessor in Wolfram's Rule 124 cellular automaton; numbers n for which A269175(n) > 0.

Original entry on oeis.org

0, 3, 6, 7, 11, 12, 14, 15, 19, 22, 24, 27, 28, 30, 31, 35, 38, 44, 47, 48, 51, 54, 55, 56, 59, 60, 62, 63, 67, 70, 76, 79, 88, 91, 94, 95, 96, 99, 102, 103, 107, 108, 110, 111, 112, 115, 118, 119, 120, 123, 124, 126, 127, 131, 134, 140, 143, 152, 155, 158, 159, 176, 179, 182, 183, 187, 188, 190, 191, 192, 195

Offset: 0

Antti Karttunen, Feb 22 2016

nonn

Sequence A269174 sorted into ascending order with duplicates removed.

The indexing starts from zero, because a(0) = 0 is a special case in this sequence. (Zero is the only number which is its own predecessor).

Antti Karttunen, Table of n, a(n) for n = 0..8940
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A269174, A269175.
Cf. A269176 (complement).
Cf. A269178 (a subsequence).

A269178 Numbers that have a unique finite predecessor in Wolfram's Rule 124 cellular automaton; numbers n for which A269175(n) = 1.

Original entry on oeis.org

0, 3, 6, 7, 11, 12, 14, 15, 19, 22, 24, 27, 28, 30, 35, 38, 44, 47, 48, 51, 54, 55, 56, 60, 67, 70, 76, 79, 88, 91, 94, 95, 96, 99, 102, 103, 107, 108, 110, 111, 112, 119, 120, 131, 134, 140, 143, 152, 155, 158, 159, 176, 179, 182, 183, 187, 188, 190, 191, 192, 195, 198, 199, 203, 204, 206, 207, 211, 214, 216, 219

Offset: 0

Antti Karttunen, Feb 22 2016

nonn

The indexing starts from zero, because a(0) = 0 is a special case in this sequence. (Zero is the only number which is its own predecessor).

Antti Karttunen, Table of n, a(n) for n = 0..5141

Cf. A269174, A269175.

Subsequence of A269177.

A269173 Formula for Wolfram's Rule 126 cellular automaton: a(n) = (n XOR 2n) OR (n XOR 4n).

Original entry on oeis.org

0, 7, 14, 15, 28, 31, 30, 27, 56, 63, 62, 63, 60, 63, 54, 51, 112, 119, 126, 127, 124, 127, 126, 123, 120, 127, 126, 127, 108, 111, 102, 99, 224, 231, 238, 239, 252, 255, 254, 251, 248, 255, 254, 255, 252, 255, 246, 243, 240, 247, 254, 255, 252, 255, 254, 251, 216, 223, 222, 223, 204, 207, 198, 195, 448, 455, 462

Offset: 0

Antti Karttunen, Feb 22 2016

nonn

			a(4) = (4 XOR 2*4) OR (4 XOR 4*4) = 12 OR 20 = 28. - _Indranil Ghosh_, Apr 02 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191
Eric Weisstein's World of Mathematics, Rule 126
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A003986, A003987, A048724, A048725, A057889.
Cf. A267365 (iterates starting from 1).
Cf. A269174.

#include 
int main()
{
    int n;
    for(n=0; n<=100; n++){
        printf("%d, ",(n^(2*n))|(n^(4*n)));
    }
    return 0;
} /* Indranil Ghosh, Apr 02 2017 */

Table[BitOr[BitXor[n, 2n], BitXor[n, 4n]], {n, 0, 100}] (* Indranil Ghosh, Apr 02 2017 *)

for(n=0, 100, print1(bitor(bitxor(n, 2*n), bitxor(n, 4*n)),", ")) \\ Indranil Ghosh, Apr 02 2017

print([(n^(2*n))|(n^(4*n)) for n in range(101)]) # Indranil Ghosh, Apr 02 2017

(define (A269173 n) (A003986bi (A048724 n) (A048725 n)))

a(n) = A048724(n) OR A048725(n) = (n XOR 2n) OR (n XOR 4n), where OR is a bitwise-or (A003986) and XOR is A003987.
Other identities. For all n >= 0:
a(2*n) = 2*a(n).
a(n) = A057889(a(A057889(n))). [Rule 126 is amphichiral (symmetric).]

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.

cp's OEIS Frontend

A269175 a(n) = number of distinct k for which A269174(k) = n; number of finite predecessors for pattern encoded in the binary expansion of n in Wolfram's Rule 124 cellular automaton.

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A269176 Numbers not present in A269174; indices of zeros in A269175.

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A332465 Numbers n for which A269174(sigma(n)) is equal to 2*A269174(n).

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

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A161903 Convert n into a sequence of binary digits, apply one step of the rule 110 cellular automaton, and interpret the results as a binary integer.

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A269177 Numbers that have a finite predecessor in Wolfram's Rule 124 cellular automaton; numbers n for which A269175(n) > 0.

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A269178 Numbers that have a unique finite predecessor in Wolfram's Rule 124 cellular automaton; numbers n for which A269175(n) = 1.

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A269173 Formula for Wolfram's Rule 126 cellular automaton: a(n) = (n XOR 2n) OR (n XOR 4n).

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