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Metadata 
bibkey: kimberling2025a035513
authors: Clark Kimberling
year: 2025
title: "OEIS A035513: Wythoff array read by falling antidiagonals"
doi: null
url: https://oeis.org/A035513
claim: "Conjecture: If m >= 2, then {(T(n,1), T(n,2)) mod m} has cardinality m^2."
strata_touched:
  - D5/S3/Arith/Wythoff/ColumnResiduePairs
license: citation-only
triage: anchor

OEIS A035513

The conjecture appears in the comment block signed Clark Kimberling, 4 June 2025. The block uses positive row indices and numbers the first two columns as 1 and 2; the first row begins 1,2,3,5,.... The official text interface returned revision 223, timestamp 13 May 2026 12:12:42, when retrieved on 3 October 2026. The assertion is still labelled Conjecture in that revision.

Henry Bottomley’s formula in the entry, signed 10 December 2001, is T(n,k)=F(k+1)*floor(n*phi)+F(k)*(n-1). R. J. Mathar’s formula, signed 3 September 2016, extends the columns by T(n,-1)=n-1, T(n,0)=floor(n*phi), and the Fibonacci recurrence. For positive rows, the formal proof identifies the nested floors with Bottomley’s first two expressions, floor(n*phi)+n-1 and 2*floor(n*phi)+n-1. The array definition continues these two columns by the same Fibonacci recurrence. The all-column closed formula is not a separate theorem in this module.

Kimberling, Complementary Equations and Wythoff Sequences, Journal of Integer Sequences 11 (2008), Article 08.3.3, defines the Wythoff array in Section 5 by W(n,h)=(n-1)*F(h)+floor(n*phi)*F(h+1). Its Theorem 10 identifies these columns with compound Wythoff sequences. The journal landing page and full PDF were retrieved on 2026-10-03; the displayed definition and Theorem 10 were checked in the extracted PDF text. They identify the array, rather than settle the residue conjecture proposed in 2025.

David R. Morrison’s A Stolarsky Array of Wythoff Pairs, page 134, defines the first two entries of row m as W(m,1)=floor(floor(m*phi)*phi) and W(m,2)=floor(floor(m*phi)*phi^2), with Fibonacci rows. The scanned primary PDF was retrieved on 2026-10-03 and its first page was visually checked. These are the nested-floor definitions used by the Lean array.

Verified locator

  • URL: https://oeis.org/A035513 (official OEIS entry; revision 223, Kimberling’s 4 June 2025 conjecture block; retrieved 2026-10-03).
  • https://oeis.org/search?q=id:A035513&fmt=text : revision 223, Kimberling’s conjecture block and the Bottomley and Mathar formulas; retrieved 2026-10-03.
  • https://oeis.org/history?seq=A035513 : history page retrieved with HTTP 200 on 2026-10-03; no claim of an exhaustive historical search.
  • https://cs.uwaterloo.ca/journals/JIS/VOL11/Kimberling/kimberling719a.html : journal title, author and article number; retrieved 2026-10-03.
  • https://cs.uwaterloo.ca/journals/JIS/VOL11/Kimberling/kimberling719a.pdf : Section 5 definition and Theorem 10; retrieved 2026-10-03.
  • https://web.math.ucsb.edu/~drm/papers/stolarsky.pdf : Morrison’s first-page array definition, printed page 134; retrieved 2026-10-03.

The bounded prior-work check is recorded in issue #12448. No claim of publication priority or global novelty is made.