A Logarithmic Asymptotic Bound and Combinatorial Invariants for OEIS A279317
Year: 2026
Type: preprint
Abstract: Let a(n) denote the minimal number of squares in a dissection of an n × (n+1) oblong into squares (OEIS sequence A279317). We report strong numerical evidence that a(n) = (16/9) ln n + 16/9 + O(1), replacing the currently listed asymptotic b(n) = round(n^(1/3)) + 6, which we show to be valid only for n ≤ 387. The new asymptotic is supported by 15 anchor points n_k* for k = 7, …, 21, spanning n ∈ [18, 49582]. We further identify two exact combinatorial identities empirically valid on all anchor points:(i) det(L_ff) · τ(G_k) = n(n+1), where G_k is the Smith graph of the optimal dissection, τ(G_k) is its spanning-tree number, and L_ff is the doubly-reduced Laplacian.(ii) τ(G_k) = n + 1 if and only if the dissection is a simple perfect squared rectangle (SPSR, all side lengths distinct); otherwise τ(G_k) = n. The Smith graphs exhibit N(k) = ⌊(k+3)/2⌋ nodes for k ≥ 11 (with N(10) = 7 as an exceptional case), asymptotically 4-regular planar structure, and second Laplacian eigenvalue λ₂(G_k) in the range [0.98, 1.59]. The exponential rate is b_∞ = lim_{k→∞} ln n_k* / k ∈ [0.557, 0.563], consistent with both 9/16 and ln(7/4) but indistinguishable from present data. We pose the proof of identities (i)–(ii) as open problems.
Author: Zeng, Changyuan
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212457175
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