Non-orientable fundamental surfaces in lens spaces

We give a concrete example of an infinite sequence of ( p n , q n ) -lens spaces L ( p n , q n ) with natural triangulations T ( p n , q n ) with p n tetrahedra such that L ( p n , q n ) contains a certain non-orientable closed surface which is fundamental with respect to T ( p n , q n ) and of minimal crosscap number among all closed non-orientable surfaces in L ( p n , q n ) and has n − 2 parallel sheets of normal disks of a quadrilateral type disjoint from the pair of core circles of L ( p n , q n ) . Actually, we can set p 0 = 0 , q 0 = 1 , p k + 1 = 3 p k + 2 q k and q k + 1 = p k + q k .