bibkey: scheuerle2025a381670 authors: Thomas Scheuerle year: 2025 title: “OEIS A381670, denominators of the compositional-square series” doi: null url: https://oeis.org/A381670 claim: “The function A(x) = x+(1/2)x^2-(1/16)x^4… = Sum_{k >= 0} x^kA381669(k)/a(k) satisfies the functional equation: x(A(x)+1) = A(A(x)). Conjecture: All terms are powers of two.” strata_touched:

• D5/S1/Recurrence/Residue/CompositionalSquareDyadicDenominators license: citation-only triage: anchor

OEIS A381670

Thomas Scheuerle’s entry, dated March 3, 2025, specifies the rational series and the denominator conjecture quoted above. A381669 records the reduced numerators; A381670 records the positive reduced denominators, including one for a zero coefficient. The normalization is A(0)=0 and [x]A(x)=1.

The module constructs F(x)=A(4x)/4 over the integers, with every coefficient of degree at least two even, satisfying F(F(x))=x+4xF(x). Appending c*x^n to an approximation tangent to x changes the degree-n residual by exactly 2c. The residual is divisible by four whenever the approximation minus x is divisible by two. Thus its negative half supplies an even correction. Compatible approximations give the formal solution. Undoing the scaling proves the original equation, and the same triangular identity proves uniqueness with the specified normalization.

For n at least two, 4^(n-1)*[x^n]A(x) is an even integer. Hence the reduced denominator divides 4^(n-1), a power of two. The constant and linear coefficients have denominator one, so the conclusion holds at every index.

Verified locator

• URL: https://oeis.org/A381670