bibkey: hanna2025a380558 authors: Paul D. Hanna year: 2025 title: “OEIS A380558, g.f. satisfying A(x - A(x)) = x^2/(1 - x^2)” doi: null url: https://oeis.org/A380558 claim: “Conjecture: a(n) is odd iff n = 2*A004760(k) for some k > 1, where A004760 lists numbers whose binary expansion does not begin 10.” strata_touched:

• D5/S1/Recurrence/Parity/A380558 license: citation-only triage: anchor

OEIS A380558

Paul D. Hanna’s February 2025 entry specifies the integral formal series A(x) = Sum_{n>=2} a(n)*x^n by A(x - A(x)) = x^2/(1 - x^2). Its formula field also gives A(x) = B(x)^2/(1 - B(x)^2), where B(x - A(x)) = x. These source formulas are literature-attested; the entry’s parity line is explicitly marked conjectural in revision 12, still live on 2026-09-28.

The linked A004760 has corrected offset one and begins 0,1,3,6,7,.... Its positive values after 1 are exactly the integers in [3*2^j, 4*2^j) for some natural j. Thus k>1 in A380558 retains the exceptional value 1, yielding n=2, and the other odd coefficients occur at twice those interval values. The upper endpoint is excluded.

Source snapshot

Retrieved from the OEIS plain-text endpoints on 2026-09-28. The SHA-256 values cover each complete response, including its header and all fields:

• A380558 revision 12 (#12 Feb 15 2025 11:16:44): 977b478467d5a09c456c0ce47539f2a3ccd2be8b6772a2a79498f78dde26d9ff
• A004760 revision 64 (#64 Jul 27 2023 08:19:18): a0c8c439079837638f02a607d722efe1a429514f53c8c0f6ce11737f000a19f8

The source fields relevant to the formal statement read:

%N A380558 G.f. A(x) satisfies A(x - A(x)) = x^2/(1 - x^2).
%C A380558 Conjecture: a(n) is odd iff n = 2*A004760(k) for some k > 1, where A004760 lists numbers whose binary expansion does not begin 10.
%F A380558 (1) A(x - A(x)) = x^2/(1 - x^2).
%F A380558 (2) A(x) = B(x)^2/(1 - B(x)^2) where B(x) = x + A(B(x)) and B(x - A(x)) = x.
%O A380558 2,2
%N A004760 List of numbers whose binary expansion does not begin 10.
%F A004760 a(1)=0, a(2)=1, a(2^m+k+2) = 2^(m+1) + 2^m+k, m >= 0, 0 <= k < 2^m. - _Yosu Yurramendi_, Jul 30 2016
%O A004760 1,3
%E A004760 Offset changed to 1, b-file corrected. - _N. J. A. Sloane_, Aug 07 2016

Verified locator

• URL: https://oeis.org/A380558
• A380558: https://oeis.org/search?q=id:A380558&fmt=text
• A004760: https://oeis.org/search?q=id:A004760&fmt=text