Cassini, d'Ocagne, and Vajda identities for n-step Fibonacci numbers

Arithmetic of Some Sequences Via $2$-determinants

We extend our investigation of $2$-determinants, which we defined in a previous paper. For a linear homogenous recurrence of the second order, we consider relations between different sequences satisfying the same linear homogeneous recurrence of the second order. After we prove a generalized identity of d'Ocagne, we derive, from a single identity, a number of classical identities (and their generalizations) such as d'Ocagne's, Cassini's, Catalan's and Vajda's. Along the way, the corresponding combinatorial interpretations in terms of restricted words over a finite alphabet are stated for the s…

k-波那契数的卡西尼恒等式

Cassini's identity for k-bonacci numbers

本文完整推导了k-波那契数的卡西尼恒等式，并推广了卡塔兰恒等式和瓦伊达恒等式，给出了简洁的证明。