103311.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

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

7 declarations, 1 contain sorry, 0 declared as axioms

Fibonacci transform satisfying $|a(n)| = F(n+1)$

The sequence $a(n)$ satisfies the linear recurrence relation: $$a(n) = 3a(n-1) - 4a(n-2) + 2a(n-3) - a(n-4)$$ with initial terms $a(0)=0, a(1)=1, a(2)=1, a(3)=0$. The sequence takes values in $\mathbb{Z}$.

References:

A103311
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA103311.a The sequence $a(n)$ satisfies the linear recurrence relation: $$a(n) = 3a(n-1) - 4a(n-2) + 2a(n-3) - a(n-4)$$ with initial terms $a(0)=0, a(1)=1, a(2)=1, a(3)=0$ line 44 source
Source documentation
The sequence $a(n)$ satisfies the linear recurrence relation: $$a(n) = 3a(n-1) - 4a(n-2) + 2a(n-3) - a(n-4)$$ with initial terms $a(0)=0, a(1)=1, a(2)=1, a(3)=0$. The sequence takes values in $\mathbb{Z}$.

no axioms what this means
Statement uses: Nat, Int
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℤ
| 0 => 0
| 1 => 1
| 2 => 1
| 3 => 0
| n + 4 => 3 * a (n + 3) - 4 * a (n + 2) + 2 * a (n + 1) - a n


@[category test, AMS 11]
```
Mentions: test, AMS
lemma OeisA103311.a_0 line 53 source
no axioms what this means
Statement uses: Eq, Int, OeisA103311.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.
lemma OeisA103311.a_1 line 56 source
no axioms what this means
Statement uses: Eq, Int, OeisA103311.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.
lemma OeisA103311.a_2 line 59 source
no axioms what this means
Statement uses: Eq, Int, OeisA103311.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.
lemma OeisA103311.a_3 line 62 source
no axioms what this means
Statement uses: Eq, Int, OeisA103311.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.
lemma OeisA103311.a_4 line 65 source
no axioms what this means
Statement uses: Eq, Int, OeisA103311.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 OeisA103311.a_abs_eq_fib sorry line 78 source
depends on sorry what this means
Statement uses: Nat, Exists, Eq, Int.natAbs, OeisA103311.a, Nat.fib
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.