A048725 -id:A048725 - cp's OEIS frontend

A048719 Binary expansion matches ((0)0011)(0*).

0, 3, 6, 12, 24, 48, 51, 96, 99, 102, 192, 195, 198, 204, 384, 387, 390, 396, 408, 768, 771, 774, 780, 792, 816, 819, 1536, 1539, 1542, 1548, 1560, 1584, 1587, 1632, 1635, 1638, 3072, 3075, 3078, 3084, 3096

Offset: 0

Antti Karttunen, Mar 30 1999

1-bits occur only in pairs, separated from other such pairs by at least two 0-bits.

All terms satisfy both A048727(n) = 3*n and A048725(n) = 5*n.

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Intersection of A048716 and A048717.

filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, !MatchQ[bb, {1}|{1, 0, _}|{_, 0, 1}|{_, 0, 1, 0, _}|{_, 1, 1, 1, _}|{_, 1, 0, 1, _}]];
Select[Range[0, 3096], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=n%3==0 && !bitand(n/3, 14*n/3) \\ Charles R Greathouse IV, Oct 03 2016

a(n) = 3*A048718(n).

A178731 a(n) = n XOR 5n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 4, 8, 12, 16, 28, 24, 36, 32, 36, 56, 60, 48, 76, 72, 68, 64, 68, 72, 76, 112, 124, 120, 100, 96, 100, 152, 156, 144, 140, 136, 132, 128, 132, 136, 140, 144, 156, 152, 228, 224, 228, 248, 252, 240, 204, 200, 196, 192, 196, 200, 204, 304, 316, 312, 292, 288, 292

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A048724, A178729, A048725, A178732, A178733, A178734, A178735, A178736. - Robert G. Wilson v, Jun 09 2010

[BitwiseXor(n, 5*n): n in [0..70]]; // Vincenzo Librandi, Jul 11 2017

f[n_] := BitXor[n, 5 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(n) = bitxor(n, 5*n); \\ Michel Marcus, Jul 11 2017

a(30) onwards from Robert G. Wilson v, Jun 09 2010

A178734 a(n) = n XOR 8n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 65, 90, 83, 108, 101, 126, 119, 144, 153, 130, 139, 180, 189, 166, 175, 216, 209, 202, 195, 252, 245, 238, 231, 288, 297, 306, 315, 260, 269, 278, 287, 360, 353, 378, 371, 332, 325, 350, 343, 432, 441, 418, 427, 404, 413, 390

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Cf. A048724, A178729, A048725, A178731, A178732, A178733, A178735, A178736. - Robert G. Wilson v, Jun 09 2010

f[n_] := BitXor[n, 8 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

def A178734(n): return n^ n<<3 # Chai Wah Wu, Jun 29 2022

a(30) onwards from Robert G. Wilson v, Jun 09 2010

A178732 a(n) = n XOR 6n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 7, 14, 17, 28, 27, 34, 45, 56, 63, 54, 73, 68, 67, 90, 85, 112, 119, 126, 97, 108, 107, 146, 157, 136, 143, 134, 185, 180, 179, 170, 165, 224, 231, 238, 241, 252, 251, 194, 205, 216, 223, 214, 297, 292, 291, 314, 309, 272, 279, 286, 257, 268, 267, 370, 381

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Cf. A048724, A178729, A048725, A178731, A178733, A178734, A178735, A178736. - Robert G. Wilson v, Jun 09 2010

f[n_] := BitXor[n, 6 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(30) onwards from Robert G. Wilson v, Jun 09 2010

A178733 a(n) = n XOR 7n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 6, 12, 22, 24, 38, 44, 54, 48, 54, 76, 70, 88, 86, 108, 102, 96, 102, 108, 150, 152, 134, 140, 182, 176, 182, 172, 166, 216, 214, 204, 198, 192, 198, 204, 214, 216, 294, 300, 310, 304, 310, 268, 262, 280, 278, 364, 358, 352, 358, 364, 342, 344, 326, 332, 438

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Cf. A048724, A178729, A048725, A178731, A178732, A178734, A178735, A178736. - Robert G. Wilson v, Jun 09 2010

f[n_] := BitXor[n, 7 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(30) onwards from Robert G. Wilson v, Jun 09 2010

A178735 a(n) = n XOR 9n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 8, 16, 24, 32, 40, 48, 56, 64, 88, 80, 104, 96, 120, 112, 136, 128, 136, 176, 184, 160, 168, 208, 216, 192, 248, 240, 232, 224, 280, 272, 264, 256, 264, 272, 280, 352, 360, 368, 376, 320, 344, 336, 424, 416, 440, 432, 392, 384, 392, 496, 504, 480, 488, 464

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Cf. A048724, A178729, A048725, A178731, A178732, A178733, A178734, A178736. - Robert G. Wilson v, Jun 09 2010

f[n_] := BitXor[n, 9 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(30) onwards from Robert G. Wilson v, Jun 09 2010

A178736 a(n) = n XOR 10n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 11, 22, 29, 44, 55, 58, 65, 88, 83, 110, 101, 116, 143, 130, 153, 176, 187, 166, 173, 220, 199, 202, 241, 232, 227, 286, 277, 260, 319, 306, 297, 352, 363, 374, 381, 332, 343, 346, 417, 440, 435, 398, 389, 404, 495, 482, 505, 464, 475, 454, 461, 572, 551

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

nonn

Cf. A048724, A178729, A048725, A178731, A178732, A178733, A178734, A178735. - Robert G. Wilson v, Jun 09 2010

f[n_] := BitXor[n, 10 n]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(30) onwards from Robert G. Wilson v, Jun 09 2010

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).]

A353167 Polynomials over GF(2) that are divisible by (x+1)^2 = x^2+1, encoded as binary numbers.

Original entry on oeis.org

0, 5, 10, 15, 17, 20, 27, 30, 34, 39, 40, 45, 51, 54, 57, 60, 65, 68, 75, 78, 80, 85, 90, 95, 99, 102, 105, 108, 114, 119, 120, 125, 130, 135, 136, 141, 147, 150, 153, 156, 160, 165, 170, 175, 177, 180, 187, 190, 195, 198, 201, 204, 210, 215, 216, 221, 225, 228, 235

Offset: 1

Jack Zhang, Apr 28 2022

Terms of A048725, sorted.
See also A001969 for those divisible by x+1 (and obviously the present sequence is a subsequence of that one).
From Kevin Ryde, Jul 22 2022: (Start)
Integers with an even number of 1-bits at even positions, and an even number of 1-bits at odd positions, and so all k with A355487(k) = 0.
Among four integers 4*i ..4*i+3, exactly one is a term here so that a(n) can be calculated by appending two bits to n-1 to ensure the two 1-bit counts are even, so a(n) = 4*(n-1) + A355487(n-1).
(End)

Cf. A001969, A048725.

Cf. A355487 (mod 4), A341389 (mod 2).

a(n) = n--; n<<2 + if(n,fold(bitxor,digits(n,4))); \\ Kevin Ryde, Jul 01 2022

More terms from David A. Corneth, Apr 28 2022

A309709 Number of binary digits that change when n is multiplied by 4.

Original entry on oeis.org

0, 2, 2, 4, 2, 2, 4, 4, 2, 4, 2, 4, 4, 4, 4, 4, 2, 4, 4, 6, 2, 2, 4, 4, 4, 6, 4, 6, 4, 4, 4, 4, 2, 4, 4, 6, 4, 4, 6, 6, 2, 4, 2, 4, 4, 4, 4, 4, 4, 6, 6, 8, 4, 4, 6, 6, 4, 6, 4, 6, 4, 4, 4, 4, 2, 4, 4, 6, 4, 4, 6, 6, 4, 6, 4, 6, 6, 6, 6, 6, 2, 4, 4, 6, 2, 2, 4, 4

Offset: 0

Ali Sada, Aug 14 2019

All terms are even.

			00101_2 * 100_2 = 10100_2: 2 bits changed, so a(5) = 2.

David A. Corneth, Table of n, a(n) for n = 0..9999
Joerg Arndt, Matters Computational (The Fxtbook)
Index entries for sequences related to binary expansion of n

Cf. A000120, A048725, A112627.

Cf. also A007302, A069010.

a:= n-> add(i, i=Bits[Split](Bits[Xor](n*4,n))):
seq(a(n), n=0..120);  # Alois P. Heinz, Aug 23 2019

a[n_] := Total@ IntegerDigits[BitXor[n, 4 n], 2]; Array[a, 88, 0] (* Giovanni Resta, Sep 19 2019 *)

A309709(n) = hammingweight(bitxor(n, n<<2)); \\ Antti Karttunen, Aug 22 2019

def A309709(n):
    s = ""
    while n > 0:
        s, n = str(n%2)+s,n//2
    s, s4, i, j = "00"+s, s+"00", 0, 0
    while i < len(s):
        if s[i] != s4[i]:
            j = j+1
        i = i+1
    return j # A.H.M. Smeets, Aug 23 2019

a(n) = A000120(A048725(n)). - Antti Karttunen, Aug 22 2019

a(A112627(n)) = 2*n and A112627(n) is the first position where 2*n occurs in this sequence. - David A. Corneth, Sep 19 2019

cp's OEIS Frontend

A048719 Binary expansion matches ((0)*0011)*(0*).

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A178731 a(n) = n XOR 5n, where XOR is bitwise XOR.

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A178734 a(n) = n XOR 8n, where XOR is bitwise XOR.

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A178732 a(n) = n XOR 6n, where XOR is bitwise XOR.

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A178733 a(n) = n XOR 7n, where XOR is bitwise XOR.

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A178735 a(n) = n XOR 9n, where XOR is bitwise XOR.

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A178736 a(n) = n XOR 10n, where XOR is bitwise XOR.

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Mathematica

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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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A353167 Polynomials over GF(2) that are divisible by (x+1)^2 = x^2+1, encoded as binary numbers.

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A048719 Binary expansion matches ((0)0011)(0*).