A048735 -id:A048735 - cp's OEIS frontend

A229763 a(n) = (2*n) XOR n AND n, where AND and XOR are bitwise logical operators.

0, 1, 2, 1, 4, 5, 2, 1, 8, 9, 10, 9, 4, 5, 2, 1, 16, 17, 18, 17, 20, 21, 18, 17, 8, 9, 10, 9, 4, 5, 2, 1, 32, 33, 34, 33, 36, 37, 34, 33, 40, 41, 42, 41, 36, 37, 34, 33, 16, 17, 18, 17, 20, 21, 18, 17, 8, 9, 10, 9, 4, 5, 2, 1, 64, 65, 66, 65, 68, 69, 66, 65, 72, 73, 74

Offset: 0

Alex Ratushnyak, Sep 28 2013

a(n) is the least significant 1-bit of each run of consecutive 1's in n, and everywhere else 0's. Or equivalently, clear to 0 each 1-bit which has another 1 immediately below. - Kevin Ryde, Feb 27 2021

			From _Kevin Ryde_, Feb 27 2021: (Start)
     n = 1831 = binary 11100100111
  a(n) =  289 = binary   100100001   low 1-bit each run
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 41.

Cf. A003188 (n XOR floor(n/2)).
Cf. A048724 (n XOR (n*2)).
Cf. A048735 (n AND floor(n/2)).
Cf. A213370 (n AND (n*2)).
Cf. A213064 (n XOR (n*2) AND (n*2)).
Cf. A229762 (n XOR floor(n/2) AND floor(n/2), 1-bit below each run).
Cf. A292272 (high 1-bit each run).

import Data.Bits ((.&.), xor, shiftL)
a229763 n = (shiftL n 1 `xor` n) .&. n :: Int
-- Reinhard Zumkeller, Oct 10 2013

Array[BitAnd[BitXor[2 #, #], #] &, 75, 0] (* Michael De Vlieger, Nov 03 2022 *)

a(n) = bitnegimply(n,n<<1); \\ Kevin Ryde, Feb 27 2021

for n in range(333): print (2*n ^ n) & n,

def A229763(n): return n&~(n<<1) # Chai Wah Wu, Jun 29 2022

a(n) = ((2*n) XOR n) AND n = ((2*n) AND n) XOR n.
a(2n) = 2a(n), a(2n+1) = A229762(n). - Ralf Stephan, Oct 07 2013
a(n) = n AND NOT 2n. - Chai Wah Wu, Jun 29 2022
G.f.: x/(1 - x^2) + Sum_{k>=1}(2^k*x^(2^k)/((1 - x)*(1 + x^(2^k))*(1 + x^(2^(k - 1))))). - Miles Wilson, Jan 24 2025

A350775 The balanced ternary expansion of a(n) is obtained by multiplying adjacent digits in the balanced ternary expansion of n: a(Sum_{k >= 0} t_k * 3^k) = Sum_{k >= 0} t_k * t_{k+1} * 3^k (with -1 <= t_k <= 1 for any k >= 0).

Original entry on oeis.org

0, 0, -1, 0, 1, -2, -3, -4, 0, 0, 0, 2, 3, 4, -5, -6, -7, -9, -9, -9, -13, -12, -11, 1, 0, -1, 0, 0, 0, -1, 0, 1, 7, 6, 5, 9, 9, 9, 11, 12, 13, -14, -15, -16, -18, -18, -18, -22, -21, -20, -26, -27, -28, -27, -27, -27, -28, -27, -26, -38, -39, -40, -36, -36

Offset: 0

Rémy Sigrist, Jan 15 2022

This sequence is to balanced ternary what A048735 is to binary, or what A330633 is to decimal.

			The first terms, in decimal and in balanced ternary, are:
  n   a(n)  bter(n)  bter(a(n))
  --  ----  -------  ----------
   0     0        0           0
   1     0        1           0
   2    -1       1T           T
   3     0       10           0
   4     1       11           1
   5    -2      1TT          T1
   6    -3      1T0          T0
   7    -4      1T1          TT
   8     0      10T           0
   9     0      100           0
  10     0      101           0
  11     2      11T          1T
  12     3      110          10
  13     4      111          11

Rémy Sigrist, Table of n, a(n) for n = 0..9841

Cf. A048735, A059095, A330633, A350776.

a(n) = { my (v=0, p=0, d); for (x=-1, oo, if (n==0, return (v), d=[0, 1, -1][1+n%3]; v+=p*d*3^x; n=(n-d)/3; p=d)) }

a(n) = 0 iff n belongs to A350776.

A269170 a(n) = n OR floor(n/2), where OR is bitwise-OR (A003986).

Original entry on oeis.org

0, 1, 3, 3, 6, 7, 7, 7, 12, 13, 15, 15, 14, 15, 15, 15, 24, 25, 27, 27, 30, 31, 31, 31, 28, 29, 31, 31, 30, 31, 31, 31, 48, 49, 51, 51, 54, 55, 55, 55, 60, 61, 63, 63, 62, 63, 63, 63, 56, 57, 59, 59, 62, 63, 63, 63, 60, 61, 63, 63, 62, 63, 63, 63, 96, 97, 99, 99, 102, 103, 103, 103, 108, 109, 111, 111, 110, 111

Offset: 0

Antti Karttunen, Feb 22 2016

Fibbinary numbers (A003714) give all integers n >= 0 for which a(n) = A003188(n) and also for which a(n) = A032766(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A000035, A003714, A003986.
Cf. A163617 (even bisection).
Cf. also A003188, A048735, A032766.

Table[BitOr[n, Quotient[n, 2]], {n, 0, 127}] (* Paolo Xausa, Aug 26 2025 *)

a(n) = bitor(n, n\2); \\ Michel Marcus, Feb 29 2016

def A269170(n): return n| n>>1 # Chai Wah Wu, Jun 29 2022

(define (A269170 n) (A003986bi n (/ (- n (A000035 n)) 2))) ;; Here A003986bi implements dyadic bitwise-OR operation (see A003986).

a(n) = A003986(n,(n-A000035(n))/2).
Other identities and observations. For all n >= 0:
a(2n) = A163617(n).
A003188(n) <= a(n) <= A032766(n).

cp's OEIS Frontend

A229763 a(n) = (2*n) XOR n AND n, where AND and XOR are bitwise logical operators.

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A350775 The balanced ternary expansion of a(n) is obtained by multiplying adjacent digits in the balanced ternary expansion of n: a(Sum_{k >= 0} t_k * 3^k) = Sum_{k >= 0} t_k * t_{k+1} * 3^k (with -1 <= t_k <= 1 for any k >= 0).

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A269170 a(n) = n OR floor(n/2), where OR is bitwise-OR (A003986).

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