lean_workbook_plus_21255

⚠️ 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 $a[n]$ be a sequence satisfying $a1=1$, $a[n+1] = 2a[n] + n*(1+2^n)$, n=1,2,3,...... then find $a[n]$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_21255 (a : ℕ → ℕ) (a1 : a 0 = 1) (a_rec : ∀ n, a (n + 1) = 2 * a n + n * (1 + 2 ^ n)) : ∀ n, a n = (n ^ 2 - n + 6) * 2 ^ (n - 2) - (n + 1)   :=  by sorry

The source starts at1, but the formal initial value is at0; negative exponent indexing is also truncated.

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

Proposed corrected statement

Use a₁=1 and aₙ₊₁=2aₙ+n(1+2ⁿ) for n≥1; state aₙ=((n²−n+6)2ⁿ−4n−4)/4 for n≥1, or the equivalent power-shift form only for n≥2.

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.