FormalConjectures.OEIS.«105565»

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import FormalConjecturesUtil

Indicator sequence for 5 Fibonacci numbers with n digits

$a(n) = 1$ if exactly 5 Fibonacci numbers exist with exactly $n$ digits, otherwise $0$. For the partial sums $S(n) = \sum_{k=1}^n a(k)$, it is conjectured that $\beta-2 < S(n)-\alpha n < \beta-1$, where $\alpha = \log(10)/\log(\phi) - 4$ and $\beta = \log(5)/(2\log(\phi)) - 1$.

References:

A105565

namespace OeisA105565open Nat Finset Real

The primary defining sequence a. $a(n) = 1$ if exactly 5 Fibonacci numbers exist with exactly $n$ digits, otherwise $0$. That is, $a(n) = 1$ if the number of indices $k \in \mathbb{N}$ such that $10^{n-1} \le \mathrm{fib}(k) < 10^n$ is 5.

def a (n : ℕ) : ℕ :=
  if n > 0 then
    -- $n$ is the number of digits, so $n \ge 1$.
    let lowerBound : ℕ := 10 ^ (n - 1)
    let upperBound : ℕ := 10 ^ n

    -- A safe upper bound for the index $k$.
    let maxK : ℕ := 5 * n + 10

    -- Count indices $k$ in range $[0, maxK)$ such that $\mathrm{fib}(k)$ has $n$ digits.
    let count : ℕ :=
      (filter (fun k => lowerBound ≤ Nat.fib k ∧ Nat.fib k < upperBound) (range maxK)).card

    if count = 5 then 1 else 0
  else
    0

Term theorems verifying the first few values of the sequence against the official OEIS b-file

@[category test, AMS 11]
theorem a_1 : a 1 = 0 := by⊢ a 1 = 0 decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_2 : a 2 = 1 := by⊢ a 2 = 1 decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_3 : a 3 = 1 := by⊢ a 3 = 1 decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_4 : a 4 = 0 := by⊢ a 4 = 0 decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_5 : a 5 = 1 := by⊢ a 5 = 1 decideAll goals completed! 🐙

The golden ratio $\phi = (1 + \sqrt{5})/2$.

noncomputable def phi : Real := goldenRatio

The constant $\alpha = \log(10)/\log(\phi) - 4$.

noncomputable def alphaConst : Real := Real.log 10 / Real.log phi - 4

The constant $\beta = \log(5)/(2\log(\phi)) - 1$.

noncomputable def betaConst : Real := Real.log 5 / (2 * Real.log phi) - 1

The partial sum $S(n) = \sum_{k=1}^n a(k)$.

noncomputable def s (n : ℕ) : Real :=
  (Finset.Icc 1 n).sum (fun k => (a k : Real))

Conjecture: $\beta-2 < S(n)-\alpha n < \beta-1$. The constants $\alpha$ and $\beta$ are as defined in the formula section.

Solved by OpenAI Codex, prompted by Adam Haig. A complete Lean 4 proof is linked by the formal_proof attribute below.

@[category research solved, AMS 11,
  formal_proof using lean4 at
    "https://github.com/HaigAd/formal-conjectures/blob/327b99914ce787ab41c67ba645626982e15b0124/FormalConjectures/OEIS/105565.lean#L595"]
theorem conjecture (n : ℕ) (hn : 1 ≤ n) :
    betaConst - 2 < s n - alphaConst * (n : Real) ∧
      s n - alphaConst * (n : Real) < betaConst - 1 := byn:ℕhn:1 ≤ n⊢ betaConst - 2 < s n - alphaConst * ↑n ∧ s n - alphaConst * ↑n < betaConst - 1
  sorryAll goals completed! 🐙

end OeisA105565