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).

REFERENCES:

Pickover, C. A. "Is There a Double Smoothly Undulating Integer?" In Computers, Pattern, Chaos and Beauty. New York: St. Martin's Press, 1990.

Pickover, C. A. "The Undulation of the Monks." Ch. 20 in Keys to Infinity. New York: W. H. Freeman, pp. 159-161 1995.

Sloane, N. J. A. Sequences A016073, A033619, A046075, A046076, and A046077 in "The On-Line Encyclopedia of Integer Sequences."