1. FormalConjectures
  2. OEIS
  3. «341254»
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Bounds on fractional parts $n r^2 - a(n)$ for $r = (2+\sqrt{5})/2$

$a(n) = \lfloor r \cdot \lfloor r \cdot n \rfloor \rfloor$, where $r = (2 + \sqrt{5})/2$.

References:

    A341254

    arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

namespace OeisA341254open Real

The constant $r = (2 + \sqrt{5})/2$.

noncomputable def rConst : ℝ := (2 + sqrt 5) / 2

The constant $r^2$.

noncomputable def rSq : ℝ := rConst * rConst

$a(n) = \lfloor r \cdot \lfloor r \cdot n \rfloor \rfloor$, where $r = (2 + \sqrt{5})/2$. Note: The original OEIS definition has $n$ starting at 1. We define $a(n)$ for all $\mathbb{N}$.

noncomputable def a (n : ℕ) : ℕ := let r := rConst let inner_floor : ℤ := Int.floor (r * n) (Int.floor (r * inner_floor.cast)).toNath1:⌊(2 + √5) / 2⌋ = 2h2:⌊(2 + √5) / 2 * 2⌋ = 4⊢ Int.toNat 4 = 4 All goals completed! 🐙h1:⌊(2 + √5) / 2 * 2⌋ = 4h2:⌊(2 + √5) / 2 * 4⌋ = 8⊢ Int.toNat 8 = 8 All goals completed! 🐙h1:⌊(2 + √5) / 2 * 3⌋ = 6h2:⌊(2 + √5) / 2 * 6⌋ = 12⊢ Int.toNat 12 = 12 All goals completed! 🐙h1:⌊(2 + √5) / 2 * 4⌋ = 8h2:⌊(2 + √5) / 2 * 8⌋ = 16⊢ Int.toNat 16 = 16 All goals completed! 🐙h1:⌊(2 + √5) / 2 * 5⌋ = 10h2:⌊(2 + √5) / 2 * 10⌋ = 21⊢ Int.toNat 21 = 21 All goals completed! 🐙

Conjecture: $1/4 < n r^2 - a(n) < 3$ for $n \ge 1$, where $r = (2 + \sqrt{5})/2$.

A formal proof has been found with the methods described in arxiv/2605.22763.

@[category research solved, AMS 11, formal_proof using formal_conjectures at "https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/341254.wip.lean#L188"] theorem a_bounds (n : ℕ) (hn : 1 ≤ n) : (1 / 4 : ℝ) < (n : ℝ) * rSq - (a n : ℝ) ∧ (n : ℝ) * rSq - (a n : ℝ) < 3 := n:ℕhn:1 ≤ n⊢ 1 / 4 < ↑n * rSq - ↑(a n) ∧ ↑n * rSq - ↑(a n) < 3 All goals completed! 🐙end OeisA341254