A038713 a(n) = n XOR (n-1), i.e., nim-sum of sequential pairs, written in binary.

1, 11, 1, 111, 1, 11, 1, 1111, 1, 11, 1, 111, 1, 11, 1, 11111, 1, 11, 1, 111, 1, 11, 1, 1111, 1, 11, 1, 111, 1, 11, 1, 111111, 1, 11, 1, 111, 1, 11, 1, 1111, 1, 11, 1, 111, 1, 11, 1, 11111, 1, 11, 1, 111, 1, 11, 1, 1111, 1, 11, 1, 111, 1, 11, 1, 1111111, 1, 11, 1, 111, 1

Offset: 1

Author

Henry Bottomley, May 02 2000

Keywords

• mult
• easy
• nonn

Examples

			a(6) = 11 because 110 XOR 101 = 11 base 2.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to Nim-sums

Crossrefs

Cf. A038712 translated to binary, A001511 translated to unary (repeated 1's).

Programs

Magma

[(10^(Valuation(n, 2)+1)-1)/9: n in [1..70]]; // Vincenzo Librandi, Mar 11 2013

Mathematica

Table[(10^IntegerExponent[2*n, 2] - 1)/9, {n, 100}] (* Vincenzo Librandi, Mar 11 2013 *)

PARI

a(n)=if(n<1,0, (10^(valuation(n,2)+1)-1)/9) /* Michael Somos, Apr 28 2005 */

Formula

a(n) = (10^A001511(n) - 1)/(10 - 1).

Multiplicative with a(2^e) = (10^(e+1) - 1)/9, a(p^e) = 1 if p odd.

G.f.: Sum_{k>=0} 10^k * x^(2^k) / (1 - x^(2^k)). - Ilya Gutkovskiy, Dec 15 2020

Dirichlet g.f.: zeta(s) * 2^s/(2^s - 10). - Amiram Eldar, Sep 21 2023