A048727 -id:A048727 - 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).

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

A277199 Transpose of square array A277320.

Original entry on oeis.org

1, 2, 2, 7, 4, 3, 4, 14, 6, 4, 13, 8, 9, 8, 5, 14, 26, 12, 28, 10, 6, 11, 28, 23, 16, 27, 12, 7, 8, 22, 18, 52, 20, 18, 14, 8, 25, 16, 29, 56, 57, 24, 21, 16, 9, 26, 50, 24, 44, 54, 46, 28, 56, 18, 10, 31, 52, 43, 32, 39, 36, 35, 32, 63, 20, 11, 28, 62, 46, 100, 40, 58, 42, 104, 36, 54, 22, 12, 21, 56, 33, 104, 125, 48, 49, 112, 101, 40, 49, 24, 13

Offset: 1

Antti Karttunen, Nov 01 2016

See A277320.

			The top left corner of the array:
   1,  2,  7,  4,  13,  14,  11,   8,  25,  26,  31,  28
   2,  4, 14,  8,  26,  28,  22,  16,  50,  52,  62,  56
   3,  6,  9, 12,  23,  18,  29,  24,  43,  46,  33,  36
   4,  8, 28, 16,  52,  56,  44,  32, 100, 104, 124, 112
   5, 10, 27, 20,  57,  54,  39,  40, 125, 114,  99, 108
   6, 12, 18, 24,  46,  36,  58,  48,  86,  92,  66,  72
   7, 14, 21, 28,  35,  42,  49,  56,  79,  70,  93,  84
   8, 16, 56, 32, 104, 112,  88,  64, 200, 208, 248, 224
   9, 18, 63, 36, 101, 126,  83,  72, 209, 202, 231, 252
  10, 20, 54, 40, 114, 108,  78,  80, 250, 228, 198, 216
  11, 22, 49, 44, 127,  98,  69,  88, 227, 254, 217, 196
  12, 24, 36, 48,  92,  72, 116,  96, 172, 184, 132, 144

Transpose: A277320.
Main diagonal: A277699.
Row 1: A065621, Row 3: 3: A277823.
Column 1: A000027, Column 3: A048727.
Cf. A048720.

(define (A277199 n) (A277320bi (A004736 n) (A002260 n))) ;; Code for A277320bi given in A277320.

A(r,c) = A048720(r, A065621(c)).

A284555 Positions of zeros in A284557.

Original entry on oeis.org

0, 3, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 24, 28, 30, 31, 33, 36, 40, 45, 48, 51, 56, 60, 62, 63, 66, 72, 77, 80, 83, 89, 90, 96, 99, 101, 102, 103, 107, 112, 115, 120, 124, 126, 127, 129, 132, 139, 144, 147, 153, 154, 160, 163, 165, 166, 167, 171, 178, 180, 187, 189, 192, 195, 197, 198, 199, 201, 202, 204, 206, 207

Offset: 0

Antti Karttunen, Apr 10 2017

Indexing starts from zero, because a(0) = 0 is a special case in this list.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A284557, A048727.

Cf. A048717 (a subsequence).

A353168 Polynomials over GF(2) that are divisible by x^2+x+1, encoded as binary numbers.

Original entry on oeis.org

0, 7, 9, 14, 18, 21, 27, 28, 35, 36, 42, 45, 49, 54, 56, 63, 65, 70, 72, 79, 83, 84, 90, 93, 98, 101, 107, 108, 112, 119, 121, 126, 130, 133, 139, 140, 144, 151, 153, 158, 161, 166, 168, 175, 179, 180, 186, 189, 195, 196, 202, 205, 209, 214, 216, 223, 224, 231, 233

Offset: 1

Jack Zhang, Apr 28 2022

nonn

Terms of A048727, sorted.

Cf. A048727.

See also A001969 for those divisible by x + 1.

from itertools import count, chain, islice
def A353168_gen(): # generator of terms
    return chain((0,),chain.from_iterable((sorted(n^ n<<1 ^ n<<2 for n in range(2**l,2**(l+1))) for l in count(0))))
A353168_list = list(islice(A353168_gen(),30)) # Chai Wah Wu, Jun 29 2022

More terms from David A. Corneth, Apr 28 2022

cp's OEIS Frontend

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

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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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A277199 Transpose of square array A277320.

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A284555 Positions of zeros in A284557.

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A353168 Polynomials over GF(2) that are divisible by x^2+x+1, encoded as binary numbers.

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Python

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