71532.lean`

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Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

10 declarations, 4 contain sorry, 0 declared as axioms

Alternating sum of signs of powers of $3/2$

The sequence $a(n) = -\sum_{k=1}^n (-1)^{\lfloor (3/2)^k \rfloor}$.

References:

A071532

Imports (1)

FormalConjecturesUtil

def OeisA71532.a The sequence $a(n) = -\sum_{k=1}^n (-1)^{\lfloor (3/2)^k \rfloor}$ line 36 source
Source documentation
The sequence $a(n) = -\sum_{k=1}^n (-1)^{\lfloor (3/2)^k \rfloor}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Neg.neg, Finset.sum, Finset.Icc, OfNat.ofNat, HPow.hPow, HDiv.hDiv
13 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℤ :=
…
```
theorem OeisA71532.a_1 Value of the sequence a at 1 line 41 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA71532.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by
…
```
Mentions: OeisA71532.a
theorem OeisA71532.a_2 Value of the sequence a at 2 line 46 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA71532.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 0 := by
…
```
Mentions: OeisA71532.a
theorem OeisA71532.a_3 Value of the sequence a at 3 line 51 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA71532.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 1 := by
…
```
Mentions: OeisA71532.a
theorem OeisA71532.a_4 Value of the sequence a at 4 line 56 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA71532.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 2 := by
…
```
Mentions: OeisA71532.a
theorem OeisA71532.a_5 Value of the sequence a at 5 line 61 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA71532.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 3 := by
…
```
Mentions: OeisA71532.a
theorem OeisA71532.conjecture1 sorry line 75 source
depends on sorry what this means
Statement uses: Iff, False, Nat, LT.lt, OfNat.ofNat, Int, OeisA71532.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA71532.conjecture2 For $n$ large enough, does $a(n) > \sqrt{n}$ always hold? sorry line 82 source
Source documentation
For $n$ large enough, does $a(n) > \sqrt{n}$ always hold?

depends on sorry what this means
Statement uses: Exists, Nat, LE.le, GT.gt, Real, OeisA71532.a, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 :
    ∃ N : ℕ, ∀ n : ℕ, N ≤ n → (a n : ℝ) > Real.sqrt (n : ℝ) := by
…
```
Mentions: OeisA71532.a
theorem OeisA71532.conjecture3 Conjecture: asymptotically, $a(n) \sim C \log(n)^2$ for some constant $C > 0$ sorry line 90 source
Source documentation
Conjecture: asymptotically, $a(n) \sim C \log(n)^2$ for some constant $C > 0$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Asymptotics.IsEquivalent, Nat, Filter.atTop, OeisA71532.a, HMul.hMul, HPow.hPow, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture3 :
    ∃ C : ℝ, 0 < C ∧ (fun n : ℕ ↦ (a n : ℝ)) ~[atTop] (fun n : ℕ ↦ C * Real.log (n : ℝ) ^ 2) := by
…
```
Mentions: OeisA71532.a
theorem OeisA71532.conjecture3_value Conjecture: the constant $C$ in $a(n) \sim C \log(n)^2$ is approximately $1.4$ sorry line 98 source
Source documentation
Conjecture: the constant $C$ in $a(n) \sim C \log(n)^2$ is approximately $1.4$.

depends on sorry what this means
Statement uses: Real, sorryAx, Bool.false, And, LT.lt, abs, HSub.hSub, OfScientific.ofScientific, Bool.true, Asymptotics.IsEquivalent, Nat, Filter.atTop, OeisA71532.a, HMul.hMul, HPow.hPow, Real.log, OfNat.ofNat
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture3_value :
    let C : ℝ := answer(sorry)
…
```