bibkey: hanna2020a338636 authors: Paul D. Hanna year: 2020 title: “OEIS A338636, continued-fraction g.f. with odd-square numerators” doi: null url: https://oeis.org/A338636 claim: “G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3^2x/(A(x) - 5^2x/(A(x) - 7^2x/(A(x) - 9^2x/(A(x) - …))))), a continued fraction relation. a(n) = 0 (mod 8) for n > 1 (conjecture).” strata_touched:

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OEIS A338636

Paul D. Hanna’s entry, dated November 4, 2020, specifies the continued-fraction generating equation and the mod-eight conjecture quoted above. The offset is 0,3, and the coefficient sequence starts 1, 1, 8, 272, 19480, 2353568. Only the mod-eight conjecture is addressed here.

For a normalized formal series A, the finite tail T(A,j,d) has d levels, begins with numerator (2j+1)^2x, and ends in A at index j+d. Thus T(A,j,0)=A and T(A,j,d+1)=A-(2j+1)^2x/T(A,j+1,d). All these denominators have constant coefficient one; division denotes their formal unit inverse. Increasing d preserves every coefficient through degree d. The defining equation 1=A-x/T(A,1,d) is imposed through degree d+1 for every d.

The module constructs the unique normalized integer series satisfying those finite-depth equations. Modulo eight every odd square equals one. The tails of 1+x agree with one through degree d, and contraction identifies the reduced series with 1+x. Consequently every coefficient of degree greater than one is divisible by eight.

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• URL: https://oeis.org/A338636