bibkey: sloane2014a239907
authors: N. J. A. Sloane; Mikhail Kurkov
year: 2021
title: "OEIS A239907, alternating count of runs of ones, with Kurkov's A329320 identity"
doi: null
url: https://oeis.org/A239907
claim: "Let cn(n,k) denote the number of times 11..1 (k 1's) appears in the binary representation of n; a(n) = n - cn(n,1) + cn(n,2) - cn(n,3) + cn(n,4) - ... . Conjecture: a(n) = n - A329320(n) for n >= 0 (noticed by Sequence Machine). - _Mikhail Kurkov_, Oct 13 2021"
strata_touched:
  - D5/S1/Digit/KurkovRunAlternatingCount
license: citation-only
triage: anchor

OEIS A239907: alternating overlapping one-block counts

Verified locator

The locator lines below were checked against https://oeis.org/A239907, https://oeis.org/A329320 and https://oeis.org/A035263 on 2026-09-15.

• URL: https://oeis.org/A239907
• A239907 %N (verbatim): Let cn(n,k) denote the number of times 11..1 (k 1’s) appears in the binary representation of n; a(n) = n - cn(n,1) + cn(n,2) - cn(n,3) + cn(n,4) - … .
• A239907 %F (verbatim): Conjecture: a(n) = n - A329320(n) for n >= 0 (noticed by Sequence Machine). - Mikhail Kurkov, Oct 13 2021
• A239907 %A (verbatim): N. J. A. Sloane, Apr 07 2014
• A329320 %N (verbatim), https://oeis.org/A329320: a(n) = Sum_{k=0..floor(log_2(n))} 1 - A035263(1 + floor(n/2^k)).
• A035263 %N (verbatim), https://oeis.org/A035263: Trajectory of 1 under the morphism 1->10, 0->11.

The identity was machine-noticed by Sequence Machine, and the supplied literature search found no proof in the searched literature.

Interpretation and scope

Occurrences overlap: cn(7,1)=3, cn(7,2)=2, and cn(7,3)=1, giving a(7)=5. Only Kurkov’s quoted %F identity is the conclusion at issue. At zero both finite index ranges are empty and the identity is 0=0.

For positive m, the supplied A035263 valuation characterization says A035263(m)=1 exactly when v_2(m) is even, or equivalently 1-A035263(m)=[v_2(m) odd]. Thus the A329320 sum becomes a cardinality of suffixes with odd valuation. This interpretation is not a separate formal proof from the morphism defining A035263. The supplied A329320 attribution is Mikhail Kurkov, Nov 10 2019, with [verification needed]. Historical openness outside the checked surfaces is unverified; no exhaustive literature search or mathematical priority is claimed.