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In this short paper, we compute the directional sequence entropy for of$\mathbb{Z}_+\times \mathbb{Z}$-actions generated by cellular automata and theshift map. Meanwhile, we study the directional dynamics of this system. As acorollary, we prove that there exists a sequence such that for any direction,some of the systems above have positive directional sequence entropy. Moreover,with help of mean ergodic theory for directional weak mixing systems, we obtaina result of number theory about combinatorial numbers.

Directional dynamics of $\mathbb{Z}_+\times \mathbb{Z}$-actions generated by 1D-CA and the shift map