Tribonacci partition formulas modulo m

On the intersection of two distinct $k$-generalized Fibonacci sequences

“…The basic properties of Tribonacci numbers can be found in [18], [24], [36], [38]. For recent papers, we refer the reader to [3], [19], [20], [33] and to the collection [21], [22], [23].…”

Tribonacci modulo $p^t$

“…Suppose p | a 11 and p ∤ a 21 . Then from (4.11) we have det A ≡ −14a 3 21 (mod p) and thus 14 ≡ 0 (mod p). As p = 2, we have p = 7.…”

“…What remains to be proved is that α 1 ∈ S p (T ). Now α 3 1 can be expressed in terms of α 1 and α 2 1 to derive the congruence (5ε+9) 2 (ε+1)(α 3 1 −α 2 1 −α 1 −1) ≡ −6(7ε 3 +29ε 2 +39ε+19) (mod p). Hence α 3 1 − α 2 1 − α 1 − 1 ≡ 0 (mod p) and thus α 1 ∈ S p (T ).…”

“…We will use the results obtained in this paper along with the results proved in [2] to solve a problem in combinatorics which is closely related to the modular periodicity of Tribonacci sequences. See [3].…”