Binomial Coefficients and Lucas Sequences

Lucas numbers of the form (2t/k)

“…Therefore, it seems natural to search the binomial coefficients n k in certain recurrences. Special cases of this question were handled by several authors, see, for example, [3].…”

Members of binary recurrences on lines of the Pascal triangle

“…Other particular instances of equation ( 4) above have been investigated in [3] (with a Lucas sequence (u n ) n≥0 and k either fixed or a prime number) and in [9] (for k = 3). We set forward the following:…”

“…Note that the right hand side of equation ( 6) above is a polynomial in x. In this case, one can use the methods of Corvaja and Zannier from [2] to show that equation ( 6) above has only finitely many solutions (n, x) when k ≥ 4 and that the same is true when k ∈ {2, 3} with finitely many exceptional parametric families of quadruples (a, b, c, d), all explicitly computable (see, for example, [3] and [7]).…”