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A027615
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Number of 1's when n is written in base -2.
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12
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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
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OFFSET
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0,3
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COMMENTS
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Base -2 is also called "negabinary".
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)
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 164.
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LINKS
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Eric Weisstein's World of Mathematics, Negabinary.
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FORMULA
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EXAMPLE
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A039724(7) = 11011 which has four 1's, so a(7) = 4.
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MATHEMATICA
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a[n_] := a[n] = a[Quotient[n - 1, -2]] + Mod[n, 2]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jul 23 2023 *)
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PROG
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(PARI) a(n) = if(n==0, 0, a(n\(-2))+n%2) /* Jianing Song, Oct 18 2018 */
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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