Complexity in Enumerative Combinatorics — Conjecture 13 (cf. [Ges-sem])

Definitions and mathematical statement

Let $i\in{\mathbb{C}}$ satisfy $i^2=-1$. For every integer $n\geq 0$, define the Gessel sequence by \[ b_n=2\cdot 5^n-(3+4i)^n-(3-4i)^n. \] The index range is supported by the source's displayed generating series, which starts at $n=0$. The source states that $b_n\in{\mathbb{Z}}_{\geq 0}$ for every $n\geq 0$, and that the sequence is C-recursive, meaning that it satisfies a linear recurrence with constant coefficients. It presents this sequence as an example without an easy combinatorial interpretation.

Unresolved conjectural statement. The supplied extraction of Conjecture 13 ends with a broken sentence referring to a sequence and to type $\mathrm{(C1)}$. It supplies neither the definition of this type nor a complete assertion about membership in it. Consequently, the sequence is precisely specified above, but the intended conjecture cannot be reconstructed as a definite mathematical statement from the supplied record. The displayed integrality expansion and generating-function formula are also truncated and are not reproduced as valid identities.

Short English statement

The record defines the Gessel sequence $b_n=2\cdot5^n-(3+4i)^n-(3-4i)^n$ for $n\geq0$. Its intended conjecture concerns type $\mathrm{(C1)}$, but the extraction omits both the type's definition and the complete conjectural assertion. This statement-only draft therefore requires source recovery and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $800329$, source problem code catalogue entry; collection: AMR Open Problem Lists. The supplied record attributes the problem to Pak, with proposed year 2018, and labels it open. This status is supplied registry metadata, not an independently verified literature assessment.
• Pak, Complexity Problems in Enumerative Combinatorics (2018), Conjecture 13, as identified in the supplied source list: https://arxiv.org/abs/1803.06636. The record reports extraction by arxiv_latex and access on 2026-10-16; these are supplied provenance details, not a claim of independent source access or verification.
• The source gives the attribution/context "cf. [Ges-sem]" and refers to that item for the definition of the Gessel sequence. No bibliographic identification or URL for [Ges-sem] is supplied.
• OEIS entry A250102, linked in the supplied statement: https://oeis.org/A250102. Its contents have not been independently checked for this draft.