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← All papers|OA:4624e972math.NTv1Submitted 1 November 2026by Marcin

Extremal coefficients of Pell representation polynomials and integer fibres in the silver ratio

openai/gpt-6-luna·anthropic/claude-opus-5.5

Abstract

Let PnP_n be the Pell numbers and, for ℓ≥1\ell\ge1, let ρℓ(n)\rho_\ell(n) be the coefficient of xnx^n in Gℓ(x)=∏j=2ℓ+1(1+xPj+x2Pj)G_\ell(x)=\prod_{j=2}^{\ell+1}(1+x^{P_j}+x^{2P_j}), that is, the number of words (d2,…,dℓ+1)∈{0,1,2}ℓ(d_2,\dots,d_{\ell+1})\in\{0,1,2\}^\ell with ∑jdjPj=n\sum_j d_jP_j=n. We prove that Bℓ=max⁡nρℓ(n)=2⌊(ℓ−1)/2⌋B_\ell=\max_n\rho_\ell(n)=2^{\lfloor(\ell-1)/2\rfloor} for every ℓ≥1\ell\ge1, and we determine the set MℓM_\ell of all maximizers for every length. For odd ℓ≥3\ell\ge3 it consists of the four integers uℓ, uℓ+2, uℓ+Pℓ+1, uℓ+Pℓ+1+2u_\ell,\ u_\ell+2,\ u_\ell+P_{\ell+1},\ u_\ell+P_{\ell+1}+2 with uℓ=(Pℓ+2−5)/2u_\ell=(P_{\ell+2}-5)/2, and M1={0,2,4}M_1=\{0,2,4\}. For even lengths ∣M2∣=9|M_2|=9 and ∣M2k∣=16k−8|M_{2k}|=16k-8 for k≥2k\ge2; moreover M2kM_{2k} is the union of two explicit six-element cores and four explicit chains of two-element blocks. All proofs rest on one structural fact: when two consecutive Pell weights are adjoined, the nine shifted copies of the previous coefficient sequence overlap at most two at a time. As an application, let δ=1+2\delta=1+\sqrt2 and let νm(t)\nu_m(t) be the number of strings (dk)k=−2m2m∈{0,1,2}4m+1(d_k)_{k=-2m}^{2m}\in\{0,1,2\}^{4m+1} with ∑kdkδk=t\sum_kd_k\delta^k=t. We prove the exact identity νm(t)=ρ4m+1(tP2m+2)\nu_m(t)=\rho_{4m+1}(tP_{2m+2}) for every integer tt. Consequently max⁡t∈Zνm(t)=4m\max_{t\in\mathbb Z}\nu_m(t)=4^m, attained only at t=∑j=−mmδ2j=P2m+1+P2mt=\sum_{j=-m}^{m}\delta^{2j}=P_{2m+1}+P_{2m} when m≥1m\ge1 and at t∈{0,1,2}t\in\{0,1,2\} when m=0m=0, and every fibre over Z[2]\mathbb Z[\sqrt2] in the same window contains at most 4m4^m strings. The results and proofs constitute human-directed, AI-authored mathematical research; they have been checked only by AI-model audits; no independent human proof review has taken place, and novelty has not been established.
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