A163617 -id:A163617 - cp's OEIS frontend

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

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)).

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

A269161 Formula for Wolfram's Rule 86 cellular automaton: a(n) = 4n XOR (2n OR n).

Original entry on oeis.org

0, 7, 14, 11, 28, 27, 22, 19, 56, 63, 54, 51, 44, 43, 38, 35, 112, 119, 126, 123, 108, 107, 102, 99, 88, 95, 86, 83, 76, 75, 70, 67, 224, 231, 238, 235, 252, 251, 246, 243, 216, 223, 214, 211, 204, 203, 198, 195, 176, 183, 190, 187, 172, 171, 166, 163, 152, 159, 150, 147, 140, 139, 134, 131, 448, 455, 462, 459

Offset: 0

Antti Karttunen, Feb 20 2016

nonn

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) = A269160(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, A163617.
Cf. A265281 (iterates starting from 1).
Cf. also A048727, A269160.

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

def A269161(n): return n<<2 ^ (n<<1 |n) # Chai Wah Wu, Jun 29 2022

(define (A269161 n) (A003987bi (* 4 n) (A003986bi (* 2 n) n))) ;; Where A003986bi and A003987bi are implementation of dyadic functions giving bitwise-OR (A003986) and bitwise-XOR (A003987) of their arguments.

a(n) = 4n XOR (2n OR n) = A003987(4*n, A003986(2*n, n)).
a(n) = 4*n XOR A163617(n).
Other identities. For all n >= 0:
a(2*n) = 2*a(n).
a(n) = A057889(A269160(A057889(n))). [Rule 86 is the mirror image of rule 30.]

A178891 a(n) = n OR 4n, where OR is bitwise OR.

Original entry on oeis.org

0, 5, 10, 15, 20, 21, 30, 31, 40, 45, 42, 47, 60, 61, 62, 63, 80, 85, 90, 95, 84, 85, 94, 95, 120, 125, 122, 127, 124, 125, 126, 127, 160, 165, 170, 175, 180, 181, 190, 191, 168, 173, 170, 175, 188, 189, 190, 191, 240, 245, 250, 255, 244, 245, 254, 255, 248, 253

Offset: 0

Dmitry Kamenetsky, Jun 21 2010

nonn

Perhaps this is a rearrangement of A115772?

Cf. A163617, A178890, A178892, A178893, A178894, A178895, A178896, A178897.

read("transforms") ; for n from 0 to 120 do printf("%d,", ORnos(n,4*n) ) ; end do: # R. J. Mathar, Jun 26 2010

f[n_] := BitOr[n, 4n]; Array[f, 58, 0] (* Robert G. Wilson v, Jun 28 2010 *)

More terms from R. J. Mathar and Robert G. Wilson v, Jun 26 2010

A178894 a(n) = n OR 7n, where OR is bitwise OR.

Original entry on oeis.org

0, 7, 14, 23, 28, 39, 46, 55, 56, 63, 78, 79, 92, 95, 110, 111, 112, 119, 126, 151, 156, 151, 158, 183, 184, 191, 190, 191, 220, 223, 222, 223, 224, 231, 238, 247, 252, 295, 302, 311, 312, 319, 302, 303, 316, 319, 366, 367, 368, 375, 382, 375, 380, 375, 382

Offset: 0

Dmitry Kamenetsky, Jun 21 2010

From Robert Israel, Dec 27 2016: (Start)
7*n <= a(n) < 8*n.
a(n) = 7*n if and only if n is in A048715.
It appears that a(n) = 8*n-1 if and only if n = (4*8^j+2*8^k+3)/7 for some j and k. (End)

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A048715, A163617, A178890, A178891, A178892, A178893, A178895, A178896, A178897.

seq(Bits:-Or(n,7*n), n=0..100); # Robert Israel, Dec 27 2016

f[n_] := BitOr[n, 7n]; Array[f, 55, 0] (* Robert G. Wilson v, Jun 28 2010 *)

More terms from Robert G. Wilson v, Jun 28 2010

A178895 a(n) = n OR 8n, where OR is bitwise OR.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 73, 90, 91, 108, 109, 126, 127, 144, 153, 146, 155, 180, 189, 182, 191, 216, 217, 218, 219, 252, 253, 254, 255, 288, 297, 306, 315, 292, 301, 310, 319, 360, 361, 378, 379, 364, 365, 382, 383, 432, 441, 434, 443, 436, 445, 438

Offset: 0

Dmitry Kamenetsky, Jun 21 2010

Perhaps this is a rearrangement of A114386?

No, e.g., a(513) = 4617 is not in A114386. Moreover, this sequence is not injective, as a(65) = a(73) = 585. - Robert Israel, Jan 31 2021

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A163617, A178890, A178891, A178892, A178893, A178894, A178896, A178897.

read("transforms"); for n from 0 to 120 do printf("%d,", ORnos(n,8*n) ) ; end do: # R. J. Mathar, Jun 26 2010

f[n_] := BitOr[n, 8n]; Array[f, 55, 0] (* Robert G. Wilson v, Jun 28 2010 *)

More terms from R. J. Mathar and Robert G. Wilson v, Jun 26 2010

A178890 a(n) = n OR 3n, where OR is bitwise OR.

Original entry on oeis.org

0, 3, 6, 11, 12, 15, 22, 23, 24, 27, 30, 43, 44, 47, 46, 47, 48, 51, 54, 59, 60, 63, 86, 87, 88, 91, 94, 91, 92, 95, 94, 95, 96, 99, 102, 107, 108, 111, 118, 119, 120, 123, 126, 171, 172, 175, 174, 175, 176, 179, 182, 187, 188, 191, 182, 183, 184, 187, 190, 187, 188

Offset: 0

Dmitry Kamenetsky, Jun 21 2010

nonn

Cf. A163617, A178891, A178892, A178893, A178894, A178895, A178896, A178897.

f[n_] := BitOr[n, 3n]; Array[f, 61, 0] (* Robert G. Wilson v, Jun 28 2010 *)

More terms from Robert G. Wilson v, Jun 28 2010

A178892 a(n) = n OR 5n, where OR is bitwise OR.

Original entry on oeis.org

0, 5, 10, 15, 20, 29, 30, 39, 40, 45, 58, 63, 60, 77, 78, 79, 80, 85, 90, 95, 116, 125, 126, 119, 120, 125, 154, 159, 156, 157, 158, 159, 160, 165, 170, 175, 180, 189, 190, 231, 232, 237, 250, 255, 252, 237, 238, 239, 240, 245, 250, 255, 308, 317, 318, 311, 312

Offset: 0

Dmitry Kamenetsky, Jun 21 2010

nonn

Cf. A163617, A178890, A178891, A178893, A178894, A178895, A178896, A178897.

f[n_] := BitOr[n, 5n]; Array[f, 57, 0] (* Robert G. Wilson v, Jun 28 2010 *)

More terms from Robert G. Wilson v, Jun 28 2010

A178893 a(n) = n OR 6n, where OR is bitwise OR.

Original entry on oeis.org

0, 7, 14, 19, 28, 31, 38, 47, 56, 63, 62, 75, 76, 79, 94, 95, 112, 119, 126, 115, 124, 127, 150, 159, 152, 159, 158, 187, 188, 191, 190, 191, 224, 231, 238, 243, 252, 255, 230, 239, 248, 255, 254, 299, 300, 303, 318, 319, 304, 311, 318, 307, 316, 319, 374, 383

Offset: 0

Dmitry Kamenetsky, Jun 21 2010

nonn

Cf. A163617, A178890, A178891, A178892, A178894, A178895, A178896, A178897.

f[n_] := BitOr[n, 6n]; Array[f, 56, 0] (* Robert G. Wilson v, Jun 28 2010 *)

More terms from Robert G. Wilson v, Jun 28 2010

A178896 a(n) = n OR 9n, where OR is bitwise OR.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 89, 90, 107, 108, 125, 126, 143, 144, 153, 178, 187, 180, 189, 214, 223, 216, 249, 250, 251, 252, 285, 286, 287, 288, 297, 306, 315, 356, 365, 374, 383, 360, 377, 378, 427, 428, 445, 446, 431, 432, 441, 498, 507, 500, 509, 502

Offset: 0

Dmitry Kamenetsky, Jun 21 2010

nonn

Cf. A163617, A178890, A178891, A178892, A178893, A178894, A178895, A178897.

f[n_] := BitOr[n, 9n]; Array[f, 55, 0] (* Robert G. Wilson v, Jun 28 2010 *)

More terms from Robert G. Wilson v, Jun 28 2010

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Go

Mathematica

Python

Scheme

Formula

A269161 Formula for Wolfram's Rule 86 cellular automaton: a(n) = 4n XOR (2n OR n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

A178891 a(n) = n OR 4n, where OR is bitwise OR.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Extensions

A178894 a(n) = n OR 7n, where OR is bitwise OR.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A178895 a(n) = n OR 8n, where OR is bitwise OR.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A178890 a(n) = n OR 3n, where OR is bitwise OR.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A178892 a(n) = n OR 5n, where OR is bitwise OR.

Views

Author

Keywords

Crossrefs