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

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import FormalConjecturesUtil

/-!
# 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](https://oeis.org/A141057)-/

namespace OeisA141057

/-- Number of Abelian cubes of length $3n$ over an alphabet of size 3. -/
def a (n : ℕ) : ℕ :=
  ∑ k ∈ Finset.range (n + 1), (n.choose k ^ 3) * ∑ j ∈ Finset.range (k + 1), (k.choose j ^ 3)

/-- Value of the sequence `a` at 0. -/
@[category test, AMS 11]
theorem a_0 : a 0 = 1 := by decide

/-- Value of the sequence `a` at 1. -/
@[category test, AMS 11]
theorem a_1 : a 1 = 3 := by decide

/-- Value of the sequence `a` at 2. -/
@[category test, AMS 11]
theorem a_2 : a 2 = 27 := by decide

/-- Value of the sequence `a` at 3. -/
@[category test, AMS 11]
theorem a_3 : a 3 = 381 := by decide

/-- Value of the sequence `a` at 4. -/
@[category test, AMS 11]
theorem a_4 : a 4 = 6219 := by decide

/--
Conjecture: the supercongruences $a(n \cdot p^k) \equiv a(n \cdot p^{k-1}) \pmod{p^{3k}}$ hold
for primes $p \ge 5$ and positive integers $n$ and $k$.-/
@[category research open, AMS 11]
theorem conjecture1 (p k n : ℕ) (hp : p.Prime) (h_p_ge_5 : 5 ≤ p) (h_k_pos : 1 ≤ k)
    (h_n_pos : 1 ≤ n) :
    (a (n * p ^ k) : ℤ) ≡ a (n * p ^ (k - 1)) [ZMOD (p ^ (3 * k))] := by
  sorry

/-- 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 $\binom{n}{k} = (-1)^k \binom{-n+k-1}{k}$. -/
def aInt (n : ℤ) : ℤ :=
  if 0 ≤ n then
    let nNat := n.toNat
    (∑ k ∈ Finset.range (nNat + 1), (nNat.choose k ^ 3) * ∑ j ∈ Finset.range (k + 1), (k.choose j ^ 3) : ℕ)
  else
    let nNat := (-n).toNat
    ∑ k ∈ Finset.range (nNat + 1), ((-1 : ℤ) ^ k * (nNat + k - 1).choose k : ℤ) ^ 3 *
      ∑ j ∈ Finset.range (k + 1), ((k.choose j : ℤ) ^ 3)

/-- Value of the sequence `aInt` at -1. -/
@[category test, AMS 11]
theorem aInt_neg_1 : aInt (-1) = -1 := by decide

/-- Value of the sequence `aInt` at -2. -/
@[category test, AMS 11]
theorem aInt_neg_2 : aInt (-2) = 255 := by decide

/--
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 $k \ge 1$.
-/
@[category research open, AMS 11]
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
  sorry

end OeisA141057

• Formalization source ↗

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.