A number of the form , , etc. The first few nontrivial undulants (with the stipulation that ) are 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, ... (OEIS A046075). Including the trivial 1- and 2-digit undulants and dropping the requirement that gives OEIS A033619.
The first few undulating squares are 121, 484, 676, 69696, ... (OEIS A016073), with no larger such numbers of fewer than a million digits (Pickover 1995). Several tricks can be used to speed the search for square undulating numbers, especially by examining the possible patterns of ending digits. For example, the only possible sets of four trailing digits for undulating squares are 0404, 1616, 2121, 2929, 3636, 6161, 6464, 6969, 8484, and 9696.
The only undulating power for and up to 100 digits is (Pickover 1995). A large undulating prime is given by (Pickover 1995).
A binary undulant is a power of 2 whose base-10 representation contains one or both of the sequences and . The first few are for , 107, 138, 159, 179, 187, 192, 199, 205, ... (OEIS A046076). The smallest for which an undulating sequence of exactly -digit occurs for , 4, ... are , 138, 875, 949, 6617, 1802, 14545, ... (OEIS A046077). An undulating binary sequence of length 10 occurs for (Pickover 1995).