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

0,3

Comments

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

Links

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.

Formula

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

Example

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

Mathematica

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

Prog

(Python)
for n in range(333): print (2*n ^ n) & n,
(Python)
def A229763(n): return n&~(n<<1) # Chai Wah Wu, Jun 29 2022
(Haskell)
import Data.Bits ((.&.), xor, shiftL)
a229763 n = (shiftL n 1 `xor` n) .&. n :: Int
-- Reinhard Zumkeller, Oct 10 2013
(PARI) a(n) = bitnegimply(n, n<<1); \\ Kevin Ryde, Feb 27 2021

Crossrefs

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).

Sequence in context: A292895 A371015 A090077 * A163509 A161399 A318479

Adjacent sequences: A229760 A229761 A229762 * A229764 A229765 A229766

Keyword

nonn,base,easy,look,changed

Author

Alex Ratushnyak, Sep 28 2013

Status

approved