bibkey: huber2023a134492 authors: Felix Huber; Artur Jasinski year: 2023 title: “OEIS A134492, a(n) = Fibonacci(6n)” doi: null url: https://oeis.org/A134492 claim: “%N a(n) = Fibonacci(6n). %C Conjecture: For n >= 2, the terms of this sequence are exactly those Fibonacci numbers which are the sum of the three numbers of a Pythagorean triple (checked up to F(80)). - Felix Huber, Nov 03 2023 %C All terms are divisible by 8. - Alonso del Arte, Jul 27 2013 %A Artur Jasinski, Oct 28 2007” strata_touched:

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

OEIS A134492

The entry is the sequence of Fibonacci numbers at indices divisible by six, 0, 8, 144, 2584, 46368, 832040, ..., with offset zero. Felix Huber’s 2023 comment characterises the terms from the second onwards as exactly those Fibonacci numbers which are perimeters of Pythagorean triangles.

Verified locator

• URL: https://oeis.org/A134492
• Locator: COMMENTS, “Conjecture: For n >= 2, the terms of this sequence are exactly those Fibonacci numbers which are the sum of the three numbers of a Pythagorean triple (checked up to F(80)). - Felix Huber, Nov 03 2023”
• Revision read: #62, Sep 22 2025. The comment is present and carries no answer.

Reading of the statement

The comment says “a Pythagorean triple” with no qualifier, so the triples range over all triples of positive integers a, b, c with a^2 + b^2 = c^2, primitive or not. The three numbers are the two legs and the hypotenuse, so their sum is the perimeter of the corresponding triangle.

Every such perimeter is even: writing a triple as d times a primitive one with parameters m > n >= 1, gcd(m, n) = 1, m - n odd, the perimeter is 2 d m (m + n). Consequently a Fibonacci number at an index not divisible by three, being odd, is never a perimeter, and those indices agree with the comment for trivial reasons.

Scope of the recorded answer

The comment is false, and the recorded counterexample is the Fibonacci number at index forty-five. That index is not a multiple of six, and the number is the perimeter of the triangle 109441 * (3145, 2928, 4297), which comes from the primitive parameters m = 61, n = 24.

It is the smallest counterexample: at indices not divisible by three the Fibonacci numbers are odd, and at indices 3, 6, ..., 42 divisible by three the comment holds. Further failures of the same kind occur at indices 57, 63, 69, 75, 81.

The counterexample lies inside the range the comment reports having checked. Under the alternative reading that restricts the triples to primitive ones the comment also fails inside that range, at indices 18 and 54, where the terms of the sequence are not primitive perimeters, and at index 81.

Bounded prior-resolution evidence

The entry was read in full at revision #62 and records no counterexample and no proof; the conjecture stands as posted. A web pass over the sequence number together with the terms of the conjecture returned no counterexample. Citation indices were not exhaustively reachable, so this is a bounded negative finding and no worldwide priority claim is made.