bibkey: fried2025proofs authors: Sela Fried year: 2025 title: “Proofs of Several Conjectures From the OEIS” doi: null url: https://cs.uwaterloo.ca/journals/JIS/VOL28/Fried/fried15.pdf claim: “The end of Section 6 conjectures that for every n in N the number F(n+2) + 2nF(n+1) is not a Fibonacci number.” strata_touched:

• D5/S3/Arith/FriedFibonacciShiftNonFibonacci
• D5/S3/Arith/GreedyFibonacciAverageClosedForm license: citation-only triage: anchor

Fried, proofs of several conjectures from the OEIS

Verified locator

DOI: none assigned

URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Fried/fried15.pdf

Journal of Integer Sequences 28 (2025), Article 25.4.3. The author is Sela Fried, Department of Computer Science, Israel Academic College in Ramat Gan. The abstract page at https://cs.uwaterloo.ca/journals/JIS/VOL28/Fried/fried15.html carries the same author, title and article number.

Definitions

The article writes F for the Fibonacci numbers with F(1) = F(2) = 1, the indexing used throughout its Section 6. Sequence A248982 is defined there as the sequence of distinct least positive numbers such that the average of the first n terms is a Fibonacci number; the article writes (a_n) for it, indexed from one.

The statement in question

Section 6 refines the closed-form conjecture recorded at A248982 to the assertion that, for n at least ten,

a_n = nF(n/2 + 3) − (n − 1)F(n/2 + 2) if n is even, and a_n = F((n+1)/2 + 2) otherwise.

It then states:

A proof of this likely proceeds along similar lines as the previous theorem. Nevertheless, we were not able to show that the two sets { nF(n/2 + 3) − (n − 1)F(n/2 + 2) : n ≥ 1 is even }, { F((n+1)/2 + 2) : n ≥ 1 is odd } are disjoint, or, equivalently, that for every n ∈ N, the number F(n + 2) + 2nF(n + 1) is not a Fibonacci number. We conjecture that this is so. Notice that a similar sequence is the Les Marvin sequence A007502(n) = F(n) + (n − 1)F(n − 1).

Both displayed sets are indexed by n ≥ 1, so the N of the equivalent form starts at one. That is forced: at n = 0 the number F(2) + 0 equals one, which is a Fibonacci number, while the two sets are unaffected.

The equivalence recorded by the source

Writing the even index as n = 2j turns the first set into the values

2jF(j + 3) − (2j − 1)F(j + 2) = 2j(F(j + 3) − F(j + 2)) + F(j + 2) = F(j + 2) + 2jF(j + 1),

which is the number named in the equivalent form. Writing the odd index as n = 2i − 1 turns the second set into {F(i + 2) : i ≥ 1}. Every value of the first set is at least four, so it can equal a Fibonacci number only at an index of at least five, and such an index lies in the second set; the disjointness of the two sets and the equivalent form therefore say the same thing.

Scope of the recorded answer

The recorded theorem D5/S3/Arith/FriedFibonacciShiftNonFibonacci.result proves the equivalent form as stated: for every n at least one and every index m, F(m) differs from F(n + 2) + 2nF(n + 1). The closed-form assertion for (a_n) itself is a different statement and is not settled here; disjointness of the two sets is a necessary ingredient of it, not the whole of it.

Bounded prior-resolution evidence

The only text claiming to settle this conjecture was arXiv:2509.26138 by Duc Hieu Le, whose Theorem 23 and Proposition 24 addressed it. That preprint was withdrawn by its author on 2025-11-06; the arXiv abstract page carries the comment “This paper has been withdrawn by Duc Hieu Le” with the author’s note that the proofs in it were machine generated and some were found to be incorrect. A literature pass opened the journal article and its abstract page, the withdrawn preprint and its listing history, the OEIS entries A248982 and A007502, and the Fibonacci literature on when a linear combination of consecutive Fibonacci numbers is again a Fibonacci number. None of them records a proof of the conjecture. Major citation-index result pages were not reachable, so this is a bounded negative finding and no worldwide priority claim is made.