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

Author

Alex Ratushnyak, Sep 28 2013

Keywords

• nonn
• base
• easy
• look

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

Examples

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

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.

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).

Programs

Haskell

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

Mathematica

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

PARI

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

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

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