87455.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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

9 declarations, 1 contain sorry, 0 declared as axioms

Expansion of $(1 - x)/(1 - 2 x + 3 x^2)$

This sequence is the expansion of $(1 - x)/(1 - 2 x + 3 x^2)$ in powers of $x$. It satisfies the linear recurrence relation $a(n) = 2 a(n-1) - 3 a(n-2)$ for $n \ge 2$, with initial values $a(0)=1$ and $a(1)=1$.

References:

A087455

Imports (1)

FormalConjecturesUtil

def OeisA87455.a The primary defining sequence a, satisfying $a(n) = 2 a(n-1) - 3 a(n-2)$ for $n \ge 2$, with initial values $a(0)=1$ and $a(1)=1$ line 36 source
Source documentation
The primary defining sequence a, satisfying $a(n) = 2 a(n-1) - 3 a(n-2)$ for $n \ge 2$, with initial values $a(0)=1$ and $a(1)=1$.

no axioms what this means
Statement uses: Nat, Int
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℤ
  | 0 => 1
  | 1 => 1
  | n + 2 => 2 * a (n + 1) - 3 * a n

/-- The decimal significand of an integer `x`: the unique real number $S(x) \in [1, 10)$ such that
$|x| = S(x) \cdot 10^k$ for some integer $k$, with the convention $S(0) = 0$. -/
```
Mentions: OeisA87455.significand
def OeisA87455.significand The decimal significand of an integer x: the unique real number $S(x) \in [1, 10)$ such that $|x| = S(x) \cdot 10^k$ for some integer $k$, with the convention $S(0) = 0$ line 43 source
Source documentation
The decimal significand of an integer x: the unique real number $S(x) \in [1, 10)$ such that $|x| = S(x) \cdot 10^k$ for some integer $k$, with the convention $S(0) = 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Real
Proof uses: HDiv.hDiv, Int.natAbs, HPow.hPow, Nat, OfNat.ofNat, HSub.hSub, List.length, Nat.digits
14 instances, coercions and auxiliary constants left out.
```
noncomputable def significand (x : ℤ) : ℝ :=
…
```
def OeisA87455.SatisfiesBenford Documented declaration: SatisfiesBenford line 49 source
Source documentation
A sequence of integers satisfies Benford's law if, for every $t \in [1, 10)$, the asymptotic relative frequency of terms whose decimal significand is at most $t$ equals $\log_{10} t$ (Berger-Hill). This is strictly stronger than the law for the leading digit alone.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Real, Membership.mem, Set, Set.Ico, OfNat.ofNat, Filter.Tendsto, HDiv.hDiv, Finset.card, Finset.filter, LE.le, OeisA87455.significand, Finset.range, Filter.atTop, nhds, Real.logb
17 instances, coercions and auxiliary constants left out.
```
def SatisfiesBenford (s : ℕ → ℤ) : Prop :=
…
```
theorem OeisA87455.a_0 Value of the sequence a at 0 line 59 source
Source documentation
Value of the sequence a at 0.

no axioms what this means
Statement uses: Eq, Int, OeisA87455.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by rfl
…
```
Mentions: OeisA87455.a
theorem OeisA87455.a_1 Value of the sequence a at 1 line 63 source
Source documentation
Value of the sequence a at 1.

no axioms what this means
Statement uses: Eq, Int, OeisA87455.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 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 rfl
…
```
Mentions: OeisA87455.a
theorem OeisA87455.a_2 Value of the sequence a at 2 line 67 source
Source documentation
Value of the sequence a at 2.

no axioms what this means
Statement uses: Eq, Int, OeisA87455.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 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 = -1 := by rfl
…
```
Mentions: OeisA87455.a
theorem OeisA87455.a_3 Value of the sequence a at 3 line 71 source
Source documentation
Value of the sequence a at 3.

no axioms what this means
Statement uses: Eq, Int, OeisA87455.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 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 = -5 := by rfl
…
```
Mentions: OeisA87455.a
theorem OeisA87455.a_4 Value of the sequence a at 4 line 75 source
Source documentation
Value of the sequence a at 4.

no axioms what this means
Statement uses: Eq, Int, OeisA87455.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 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 = -7 := by rfl
…
```
Mentions: OeisA87455.a
theorem OeisA87455.conjecture It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017; Arno Berger, email, Jan 06 2017] sorry line 81 source
Source documentation
It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017; Arno Berger, email, Jan 06 2017]. - N. J. A. Sloane, Feb 08 2017

depends on sorry what this means
Statement uses: Iff, True, OeisA87455.SatisfiesBenford, OeisA87455.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture : answer(sorry) ↔ SatisfiesBenford a := by
…
```
Mentions: OeisA87455.SatisfiesBenford, OeisA87455.a