A sequence [Formula: see text] over ℤ is an LP-sequence if for every prime p and integer n ≥ 0 we have [Formula: see text] (mod p), when [Formula: see text] is a base p expansion of n. In this paper, we study sequences [Formula: see text] such that both [Formula: see text], [Formula: see text] are LP-sequences for some d ≥ 2. One of those sequences is a counter-example to a conjecture of McIntosh [15].