Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p-1}\equiv (p/3)3^{p-1}\left(modp²\right)$, where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1+x+x^{-1})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0pt}{}{p-1}{k}\right)\left(\genfrac{}{}{0pt}{}{2k}{k}\right)({(-1)}^k-{(-3)}^{-k})\equiv (p/3)(3^{p-1}-1)\left(modp³\right)$. In addition, we get some new combinatorial identities.

Melham discovered the Fibonacci identity $F_{n+1}{F}_{n+2}{F}_{n+6}-F{³}_{n+3}=(-1)ⁿFₙ$. He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and $Wₙ=p{W}_{n-1}+q{W}_{n-2}$ and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: $W_{n+1}{W}_{n+2}{W}_{n+6}-W{³}_{n+3}=e{q}^{n+1}(p³W_{n+2}-q²W_{n+1})$. There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: $FₙF_{n+4}{F}_{n+5}-F{³}_{n+3}={(-1)}^{n+1}{F}_{n+6}$. We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is $WₙW_{n+4}{W}_{n+5}-W{³}_{n+3}=eqⁿ(p³W_{n+4}-q{W}_{n+5})$.

In this paper we consider two parameters generalization of the Fibonacci numbers and Pell numbers, named as the $(k,p)$-Fibonacci numbers. We give some new interpretations of these numbers. Moreover using these interpretations we prove some identities for the $(k,p)$-Fibonacci numbers.

Let $d$ be a fixed positive integer. A Lucas $d$-pseudoprime is a Lucas pseudoprime $N$ for which there exists a Lucas sequence $U(P,Q)$ such that the rank of $N$ in $U(P,Q)$ is exactly $(N-\epsilon (N\left)\right)/d$, where $\epsilon$ is the signature of $U(P,Q)$. We prove here that all but a finite number of Lucas $d$-pseudoprimes are square free. We also prove that all but a finite number of Lucas $d$-pseudoprimes are Carmichael-Lucas numbers.