141057.lean`

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

11 declarations, 2 contain sorry, 0 declared as axioms

Number of Abelian cubes of length $3n$ over an alphabet of size 3

An Abelian cube is a string of the form $x x' x''$ with $|x| = |x'| = |x''|$ and $x$ is a permutation of $x'$ and $x''$. The number of Abelian cubes of length $3n$ over an alphabet of size 3 is given by $$a(n) = \sum_{k=0}^n \binom{n}{k}^3 \sum_{j=0}^k \binom{k}{j}^3.$$

References:

A141057

Imports (1)

FormalConjecturesUtil

def OeisA141057.a Number of Abelian cubes of length $3n$ over an alphabet of size 3 line 36 source
Source documentation
Number of Abelian cubes of length $3n$ over an alphabet of size 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, HPow.hPow, Nat.choose
10 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA141057.a_0 Value of the sequence a at 0 line 41 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA141057.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 decide
…
```
Mentions: OeisA141057.a
theorem OeisA141057.a_1 Value of the sequence a at 1 line 45 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA141057.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 = 3 := by decide
…
```
Mentions: OeisA141057.a
theorem OeisA141057.a_2 Value of the sequence a at 2 line 49 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA141057.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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_2 : a 2 = 27 := by decide
…
```
Mentions: OeisA141057.a
theorem OeisA141057.a_3 Value of the sequence a at 3 line 53 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA141057.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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_3 : a 3 = 381 := by decide
…
```
Mentions: OeisA141057.a
theorem OeisA141057.a_4 Value of the sequence a at 4 line 57 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA141057.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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_4 : a 4 = 6219 := by decide
…
```
Mentions: OeisA141057.a
theorem OeisA141057.conjecture1 sorry line 64 source
depends on sorry what this means
Statement uses: Nat, Nat.Prime, LE.le, OfNat.ofNat, Int.ModEq, HPow.hPow, Int, HMul.hMul, OeisA141057.a, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def OeisA141057.aInt Extension of the sequence $a$ to all integers $n \in \mathbb{Z}$ line 72 source
Source documentation
Extension of the sequence $a$ to all integers $n \in \mathbb{Z}$. For $n < 0$, $a(n) = \sum_{k=0}^{-n} \binom{n}{k}^3 \sum_{j=0}^k \binom{k}{j}^3$, where $\binom{n}{k} = (-1)^k \binom{-n+k-1}{k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int
Proof uses: ite, LE.le, OfNat.ofNat, Nat, Int.toNat, Finset.sum, Finset.range, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat.choose, Neg.neg, HSub.hSub
21 instances, coercions and auxiliary constants left out.
```
def aInt (n : ℤ) : ℤ :=
…
```
theorem OeisA141057.aInt_neg_1 Value of the sequence aInt at -1 line 83 source
Source documentation
Value of the sequence aInt at -1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA141057.aInt, Neg.neg, OfNat.ofNat
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 aInt_neg_1 : aInt (-1) = -1 := by decide
…
```
Mentions: OeisA141057.aInt
theorem OeisA141057.aInt_neg_2 Value of the sequence aInt at -2 line 87 source
Source documentation
Value of the sequence aInt at -2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA141057.aInt, Neg.neg, OfNat.ofNat
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 aInt_neg_2 : aInt (-2) = 255 := by decide
…
```
Mentions: OeisA141057.aInt
theorem OeisA141057.conjecture2 Conjecture (Peter Bala, 2022): The supercongruences sorry line 96 source
Source documentation
Conjecture (Peter Bala, 2022): The supercongruences $a(n \cdot p^k) \equiv a(n \cdot p^{k-1}) \pmod{p^{3k}}$ hold for the integer-indexed extension $a(n)$ for all $n \in \mathbb{Z} \setminus \{0\}$, primes $p \ge 5$, and $k \ge 1$.

depends on sorry what this means
Statement uses: Nat, Int, Nat.Prime, LE.le, OfNat.ofNat, Ne, Int.ModEq, HPow.hPow, HMul.hMul, OeisA141057.aInt, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 (p k : ℕ) (n : ℤ) (hp : p.Prime) (h_p_ge_5 : 5 ≤ p) (h_k_pos : 1 ≤ k)
    (hn : n ≠ 0) :
    aInt (n * (p ^ k : ℤ)) ≡ aInt (n * (p ^ (k - 1) : ℤ)) [ZMOD (p ^ (3 * k) : ℤ)] := by
…
```
Mentions: OeisA141057.aInt