An undulating number is a nonnegative integer whose decimal expansion alternates between two
digits 
and 
. The first few nontrivial undulants, which have at least three
digits and satisfy 
, are 101, 121, 131, 141, 151, 161, 171, 181, 191, 202,
212, ... (OEIS A046075). Including the trivial
1- and 2-digit undulants and allowing 
gives OEIS A033619.
The only nontrivial undulating squares are 121, 484, 676, and 69696 (OEIS A016073; Moews 1998, pp. 1-7). Pickover (1995) had reported that no larger examples have fewer than a million digits. Several tricks can be used to speed the search for undulating squares, especially by examining the possible patterns of ending digits. For example, a nontrivial undulating square number with at least four digits must end in one of 0404, 1616, 2121, 2929, 3636, 6161, 6464, 6969, 8484, and 9696.
Among nontrivial undulating numbers with at most 100 digits, the only perfect power 
with a positive integer
base 
and an integer exponent 
is 
(Pickover 1995). A large undulating prime
number is given by 
(Pickover 1995).
A binary undulant is a power of 2 whose decimal expansion contains the substring 010 or 101. The
first few are 
for 
,
107, 138, 159, 179, 187, 192, 199, 205, ... (OEIS A046076).
The smallest 
for which the longest substring of alternating digits
0 and 1 in the decimal expansion of 
has length exactly 
, for 
, 4, ..., are 103, 138, 875, 949, 6617, 1802, 14545, ...
(OEIS A046077). The longest such substring
has length 10 for 
(Pickover 1995).
