lean_workbook_plus_9944

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let the sequence $a_{n}$ : \n $\left\{\begin{matrix} a_{0}=a_{1}=1& \ a_{n+2}=[\frac{(a_{n+1})^{2}+2}{a_{n}}] \end{matrix}\right.$ \n Prove $a_{n}a_{n+2}-(a_{n+1})^{2}= 2 \forall n\geq 0$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_9944 (a : ℕ → ℝ) (a0 : a 0 = 1 ∧ a 1 = 1) (a_rec : ∀ n, a (n + 2) = (a (n + 1))^2 + 2 / a n) : ∀ n, a n * a (n + 2) - (a (n + 1))^2 = 2   :=  by sorry

The source floor of a whole quotient was replaced by a square plus 2/aₙ.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Take a₀=a₁=1 and aₙ₊₂=⌊(aₙ₊₁²+2)/aₙ⌋; prove positivity/integrality and aₙaₙ₊₂−aₙ₊₁²=2 for every n≥0.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.