Congruences for sequences similar to Euler numbers

For a≠0 we define by (n=0,1,2,…), where [n/2]=n/2 or (n−1)/2 according as 2|n or 2∤n. In the paper we establish many congruences for modulo prime powers, and show that there is a set X and a map T:X→X such that is the number of fixed points of Tn.