A027615 Number of 1's when n is written in base -2.

0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 4, 5, 3, 4, 5, 6, 4, 5, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 3, 4, 2

Offset: 0

Author

Pontus von Brömssen, Nov 14 1997

Keywords

• nonn
• base

Comments

Base -2 is also called "negabinary".

From Jianing Song, Oct 18 2018: (Start)

Define f(n) as: f(0) = 0, f(-2*n) = f(n), f(-2*n+1) = f(n) + 1, then a(n) = f(n), n >= 0. See A320642 for the other half of f.

For k > 0, the earliest occurrence of k is n = A305750(k).

Conjecture: a(n) != A053737(n) if and only if there exists even k >= 4 such that n mod 2^k >= (5*2^(k+1) + 2)/3. If this holds, then the probability of a random chosen number n to satisfy a(n) != A053737(n) is 1/6. (End)

Examples

			A039724(7) = 11011 which has four 1's, so a(7) = 4.
		

References

• J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 164.

Links

• Jianing Song, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Negabinary.
• Wikipedia, Negative base.

Crossrefs

Cf. A000120, A005351, A039724, A053737, A072894, A305750, A320642.

Programs

Mathematica

a[n_] := a[n] = a[Quotient[n - 1, -2]] + Mod[n, 2]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jul 23 2023 *)

PARI

a(n) = if(n==0, 0, a(n\(-2))+n%2) /* Jianing Song, Oct 18 2018 */

Formula

a(n) = 3*A072894(n+1) - 2*n - 3. Proof by Nikolaus Meyberg, following a conjecture by Ralf Stephan. - R. J. Mathar, Jan 11 2013

a(n) == n (mod 3). - Jianing Song, Oct 18 2018

a(n) = A000120(A005351(n)). - Michel Marcus, Oct 23 2018