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bibkey: hogan2026a400429 authors: Hunter Hogan year: 2026 title: “OEIS A400429, semi-meanders by crossings and winding number” doi: null url: https://oeis.org/A400429 claim: “Conjecture: D_2(n) = (n^2 + 2*n + n mod 2 - 20)/2, for n >= 4.” strata_touched:

  • D5/S3/Combinatorics/SemiMeanderSecondDiagonal license: citation-only triage: anchor

OEIS A400429

Hunter Hogan’s A400429, revision 16 (26 September 2026), counts connected semi-meanders by crossing number and winding. The official text retrieved from https://oeis.org/search?q=id%3AA400429&fmt=text on 29 September 2026 has SHA-256 6a96f289d90c588a6672d550008d30d36a7709e5dbecab3980d9869c4d0de600. That hash covers the search response for this one entry, including its wrapper and licence footer; it is not a hash of the OEIS revision archive or linked table. The inspected %I line reads %I A400429 #16 Sep 26 2026 16:33:24.

Verified locator

  • URL: https://oeis.org/A400429 (revision 16, name, comment and formula fields; retrieved 2026-09-29 through the official OEIS text interface).
  • Response SHA-256: 6a96f289d90c588a6672d550008d30d36a7709e5dbecab3980d9869c4d0de600. This identifies the retrieved search response, not the revision archive.

Exact source lines

%N A400429 Irregular triangle read by rows: T(n,k) is the number of semi-meanders with n crossings and winding number 2*(k-1) + (n mod 2), for n >= 2, 1 <= k <= floor(n/2).

%C A400429 Equivalent to a line with vertices, 1,…,2n, and each vertex, i, joined by an arch below the line to vertex 2n+1-i for 1 <= i <= n. T(n,k) counts loops that are the union of noncrossing perfect matchings of lower arches with 2*(k-1) + (n mod 2) upper arches that join a vertex <= n to a vertex > n.

%F A400429 Let D_j(n) = T(n,floor(n/2)-j+1) denote the j-th diagonal, for 1 <= j <= floor(n/2).

%F A400429 Conjecture: D_2(n) = (n^2 + 2*n + n mod 2 - 20)/2, for n >= 4.

The lower arches are the fixed rainbow; the upper arches form a noncrossing perfect matching. The union must be one loop. In zero-based endpoint notation, the lower partner of x is 2*n-1-x, and each upper arch crossing the midpoint has one endpoint below n and one at least n. For j=2, the source’s k=floor(n/2)-1 gives winding 2*(floor(n/2)-2)+(n mod 2)=n-4 for every n>=4.

The entry calls the displayed diagonal formula a conjecture. This note records its source wording and model, not publication priority or an independent proof.