Diophantine problems in recurrences

Squares, perfect powers, repdigits, factorials and powers of two among terms of recurrence sequences.

TheoremTribonacci numbers that are products of factorials#●ready to prove

Let \((T_n)_{n \in \mathbb{Z}}\) be the integer sequence with \(T_0 = 0\), \(T_1 = T_2 = 1\) and \(T_{j+3} = T_{j+2} + T_{j+1} + T_j\) for every integer \(j\). If \(n\) is an integer such that \(|T_n| = m_1! \, m_2! \cdots m_k!\) for some \(k \ge 1\) and some positive integers \(m_1, \dots, m_k\), then \(n \in \{-9, -8, -7, -5, -3, -2, 1, 2, 3, 4, 7\}\).

public theorem_wanted tribonacci_factorialProduct_index_mem
    (T : ℤ → ℤ)
    (hT0 : T 0 = 0)
    (hT1 : T 1 = 1)
    (hT2 : T 2 = 1)
    (hTrec : ∀ j : ℤ, T (j + 3) = T (j + 2) + T (j + 1) + T j)
    (n : ℤ)
    (k : ℕ)
    (hk : 0 < k)
    (m : Fin k → ℕ)
    (hm : ∀ i, 0 < m i)
    (h : (T n).natAbs = ∏ i, (m i).factorial) :
    n ∈ ({-9, -8, -7, -5, -3, -2, 1, 2, 3, 4, 7} : Finset ℤ)
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