bibkey: stephan2010a107928 authors: Ralf Stephan year: 2010 title: “OEIS A107928, a(n) is the numerator of harmonic mean of a(n-1) and a(n-2)” doi: null url: https://oeis.org/A107928 claim: “%N a(n) is the numerator of harmonic mean of a(n-1) and a(n-2). %F a(1)=2; a(2)=3; n>=3: a(n) = numerator(2a(n-1)a(n-2)/(a(n-2)+a(n-1))). %F Conjecture: G.f.: x(2+3x+12x^2+8x^3+8x^4)/[(1-2x)(1+2x+4x^2)]. a(3n) = (3/2)8^n, a(3n+1) = 38^n, a(3n+2) = 28^n, for n>0. - Ralf Stephan, Dec 01 2010 %A Zak Seidov, Jun 10 2005” strata_touched:

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OEIS A107928

Zak Seidov submitted the sequence on June 10, 2005. Its terms are the reduced numerators obtained by repeatedly taking the harmonic mean of the preceding two terms, starting from 2 and 3. Ralf Stephan added the three-residue closed form as a conjecture on December 1, 2010.

The repository formalizes the sequence over the natural numbers with a zero sentinel at index zero and proves the three-residue closed form for every positive block index.

Verified locator

url: https://oeis.org/A107928

doi: null (OEIS entry; no DOI assigned).

The FORMULA field at https://oeis.org/A107928/internal states, verbatim:

Conjecture: G.f.: x*(2+3x+12x^2+8x^3+8x^4)/[(1-2x)(1+2x+4x^2)]. a(3n) = (3/2)8^n, a(3n+1) = 38^n, a(3n+2) = 2*8^n, for n>0. - Ralf Stephan, Dec 01 2010

Only the three-residue closed form is formalized, by elementary block induction; no priority claim is made. The printed generating function with +8x^4 expands to 2, 3, 12, 24, 32, … and disagrees with the sequence at n = 5 (a(5) = 16), whereas changing the sign to -8x^4 yields the rational generating function x(2+3x+12x^2+8x^3-8x^4)/(1-8x^3) of the proved closed form; the generating-function sentence is neither formalized nor claimed, and the discrepancy is recorded here, not corrected upstream.