Mathar's first recurrence for OEIS A152661

OEIS A152661, revision 23, defines a(n), for n >= 2, as the number of permutations of {1,...,n} whose first two entries have the same parity. Its first conjectural FORMULA, attributed to R. J. Mathar, states

$$ (3-n)a(n)+2(n-2)a(n-1)+(n-1)(n-2)(n-3)a(n-2)=0. $$

The assertion here is this exact equality in the integers for every natural n >= 4. Each sequence value is the actual permutation cardinality, cast to the integers. The domain ensures that n, n-1 and n-2 all lie in the source domain. The adjacent conjecture a(n)=2*A077613(n-1) is excluded.

The recurrence relates three consecutive values of the source permutation count at every admissible index. Issue #22922 preregisters precisely this first Mathar FORMULA and its counting-bridge requirement.

The source already supplies even and odd factorial-binomial formulas. Those are prior formulas, not new counting claims here. The missing result targeted by #22922 is a proof of the conjectural recurrence for the actual source count. The preregistration records bounded source, repository and literature searches without a matching settlement; it does not establish global priority.

Represent a permutation by a bijection of Fin n, giving its value in position i+1 the natural label p(i).val+1. Equality of parity is preserved by adding one to both first values. Map each counted permutation to its ordered pair of distinct first values. This gives an equivalence with the disjoint union of the corresponding completion fibers.

For each allowed pair, the number of permutations completing it is (n-2)!. The proof applies the frozen finite partial-injection extension theorem D5/S3/Combinatorics/EdgeLabeling/CubicARGraphCounting.nat_card_equiv_constraints to the two prescribed positions and values. If the parity classes have sizes e and o, the pair count is e(e-1)+o(o-1). The equivalence therefore proves that the actual source cardinality is e(e-1)+o(o-1)!.

A natural n >= 4 whose actual permutation count violates the displayed integer equality would contradict the theorem. Replacing that count by an unconnected factorial formula, or checking only finitely many indices, would not establish this assertion.

The mathematical source is D5/S3/Combinatorics/Permutation/MatharSameParityRecurrence.lean. Its public definition a is the integer cast of the cardinality of the permutation subtype for n >= 2, with zero assigned only outside that domain. The public theorem D5/S3/Combinatorics/Permutation/MatharSameParityRecurrence.result proves the displayed recurrence for every natural n >= 4. Its Blueprint mirrors both public declarations and binds the recurrence to this problem.

The positive four-slot escape audit is unfinished, as documented in issue #22961. No validated registration or refutation exemption is claimed. The missing evidence is a compiled count-realization bridge for a(4)>0, actual observation dependence at indices 4 and 5, and a validated current four-slot binding.

For n=2k+2, the two parity classes both have size k+1, so the ordered-pair count is 2k(k+1) and a(2k+2)=2k(k+1)(2k)!. For n=2k+3, their sizes are k+1 and k+2, so the pair count is 2(k+1)^2 and a(2k+3)=2(k+1)^2(2k+1)!. These identities are obtained from the actual cardinality through the fiber equivalence. They recover the source's prior parity formulas; they are not additional open-problem settlements.

Substitution of these formulas and the factorial successor identity proves the recurrence separately for n=2k+4 and n=2k+5, covering every n >= 4. The first case includes n=4 and uses a(2)=0 and a(4)=8. This establishes exactly the first Mathar FORMULA. The adjacent A077613 conjecture remains unresolved in this delivery and contributes no additional solution credit.

The recorded literature searches are bounded. Worldwide novelty, first-proof priority and the absence of an independent proof outside their scope are unverified. No assertion is made about other OEIS conjectures or an extension of this recurrence below n=4.