lean_workbook_plus_47356

⚠️ 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 $a_n=[(4+\sqrt{15})^{n}+1]$. Show that $a_n=(4+\sqrt{15})^{n}+(4-\sqrt{15})^{n}$, $a_{n+2}=8a_{n+1}-a_n$, and $a_0=2, a_1=8$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_47356 (a : ℕ → ℝ) (a_n_def : ∀ n, a n = (4 + Real.sqrt 15) ^ n + 1) : ∀ n, a n = (4 + Real.sqrt 15) ^ n + (4 - Real.sqrt 15) ^ n ∧ a (n + 2) = 8 * a (n + 1) - a n ∧ a 0 = 2 ∧ a 1 = 8   :=  by sorry

The square brackets representing an integer part were dropped.

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

Proposed corrected statement

Define a_n=floor((4+√15)^n+1), then prove the stated integer-valued formula, recurrence, and initial data.

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.