100478.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

Pentanacci $\pi$ sequence

Start with $a(1)=a(2)=a(3)=a(4)=a(5)=1$ and for $n>5$, $a(n) = \pi(\sum_{j=1}^5 a(n-j))$ where $\pi = A000720$.

References:

A100478

Imports (1)

FormalConjecturesUtil

def OeisA100478.a The primary defining sequence a line 42 source
Source documentation
The primary defining sequence a. Pentanacci $\pi$ sequence: $a(1)=a(2)=a(3)=a(4)=a(5)=1$; for $n>5$, $a(n) = \pi(\sum_{j=1}^5 a(n-j))$ where $\pi = A000720$. Note on indices: for $n \ge 0$, $a(n)$ corresponds to $A_{n+1}$ in the OEIS sequence.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
def OeisA100478.aGeneral A general sequence defined by the Pentanacci $\pi$ recurrence, starting with arbitrary initial values $v: \text{Fin } 5 \to \mathbb{N}$ line 58 source
Source documentation
A general sequence defined by the Pentanacci $\pi$ recurrence, starting with arbitrary initial values $v: \text{Fin } 5 \to \mathbb{N}$. The sequence $a_{\mathrm{general}}(v, n)$ is the n-th term (0-indexed).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fin, OfNat.ofNat, Nat
4 instances, coercions and auxiliary constants left out.
```
noncomputable def aGeneral (v : Fin 5 → ℕ) (n : ℕ) : ℕ :=
…
```
theorem OeisA100478.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 73 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100478.a, OfNat.ofNat
Proof uses: Eq.refl
2 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: OeisA100478.a
theorem OeisA100478.a_1 line 76 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100478.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA100478.a_2 line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100478.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA100478.a_3 line 82 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100478.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA100478.a_4 line 85 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100478.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA100478.conjecture Starting with other values of $a(1)$, $a(2)$, $a(3)$, $a(4)$, $a(5)$ what behaviors are possible? sorry line 99 source
Source documentation
Starting with other values of $a(1)$, $a(2)$, $a(3)$, $a(4)$, $a(5)$ what behaviors are possible? Does the sequence always stick at a single integer after some point, or can it go into a loop, or is there a third pattern?
KitaKen1 showed the sequence either becomes constant or goes into a loop, see https://github.com/KitaKen1/oeis-a100478-eventual-periodicity/blob/642eed0ffee26415528ab8c48c5181826be04860/lean/OeisA100478FC.lean#L158-L168.
tadamcz showed the sequence always becomes constant, see https://tadamcz.com/fc-review-results/aea251bb26/#/f/OEIS/100478.

depends on sorry what this means
Statement uses: Iff, True, Fin, OfNat.ofNat, Nat, LT.lt, Exists, And, Membership.mem, Finset, Insert.insert, Singleton.singleton, Filter.Eventually, Eq, OeisA100478.aGeneral, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture:
  answer(True) ↔ ∀ v : Fin 5 → ℕ, (∀ i, 0 < v i) → ∃ c ∈ ({66, 67, 68, 70, 71, 72} : Finset ℕ),
  ∀ᶠ n in Filter.atTop, aGeneral v n = c := by
…
```
Mentions: c, OeisA100478.aGeneral